diff --git "a/Lemmas/imo_1985_p6_lemmas.lean" "b/Lemmas/imo_1985_p6_lemmas.lean"
new file mode 100644--- /dev/null
+++ "b/Lemmas/imo_1985_p6_lemmas.lean"
@@ -0,0 +1,14976 @@
+import Mathlib
+
+set_option linter.unusedVariables.analyzeTactics true
+
+lemma imo_1985_p6_3
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y) :
+  ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x := by
+  intros n x hx₀
+  cases' hx₀ with hn₀ hx₁
+  have g₂₀: f n 1 ≤ f n x := by
+    by_cases hx₂: 1 < x
+    . refine le_of_lt ?_
+      refine h₄ n 1 x ?_ hx₂
+      exact Nat.zero_lt_of_lt hn₀
+    . push_neg at hx₂
+      have hx₃: x = 1 := by exact le_antisymm hx₂ hx₁
+      rw [hx₃]
+  have g₂₁: f 1 1 < f n 1 := by
+    rw [h₀]
+    refine Nat.le_induction ?_ ?_ n hn₀
+    . rw [h₁ 1 1 (by norm_num), h₀]
+      norm_num
+    . intros m hm₀ hm₁
+      rw [h₁ m 1 (by linarith)]
+      refine one_lt_mul_of_lt_of_le hm₁ ?_
+      norm_cast
+      nth_rw 1 [← add_zero 1]
+      refine add_le_add ?_ ?_
+      . exact le_of_lt hm₁
+      . refine one_div_nonneg.mpr ?_
+        exact Nat.cast_nonneg' m
+  refine lt_of_lt_of_le ?_ g₂₀
+  exact (lt_iff_lt_of_cmp_eq_cmp (congrFun (congrArg cmp (h₀ 1)) (f n 1))).mp g₂₁
+
+lemma imo_1985_p6_6_3_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x):
+   Continuous (f (Nat.succ 0)) := by
+  have hn₁: f 1 = fun (x:NNReal) => (x:ℝ) := by exact (Set.eqOn_univ (f 1) fun x => ↑x).mp fun ⦃x⦄ _ => h₀ x
+  rw [hn₁]
+  exact NNReal.continuous_coe
+
+
+lemma imo_1985_p6_6_4
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n)):
+   ∀ (n : ℕ), Nat.succ 0 ≤ n → Continuous (f n) → Continuous (f (n + 1)) := by
+  intros d hd₀ hd₁
+  have hd₂: f (d + 1) = fun x => f d x * (f d x + 1 / ↑d) := by
+    exact (Set.eqOn_univ (f (d + 1)) fun x => f d x * (f d x + 1 / ↑d)).mp fun ⦃x⦄ _ => h₁ d x hd₀
+  rw [hd₂]
+  refine Continuous.mul hd₁ ?_
+  refine Continuous.add hd₁ ?_
+  exact continuous_const
+
+
+lemma imo_1985_p6_6_5
+  (f : ℕ → NNReal → ℝ)
+  (d : ℕ)
+  (hd₁ : Continuous (f d))
+  (hd₂ : f (d + 1) = fun x => f d x * (f d x + 1 / ↑d)):
+   Continuous (f (d + 1)) := by
+  rw [hd₂]
+  refine Continuous.mul hd₁ ?_
+  refine Continuous.add hd₁ ?_
+  exact continuous_const
+
+
+
+
+lemma imo_1985_p6_7
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (hmo₄ : ∀ (n : ℕ), 0 < n → Continuous (f₀ n)) :
+  ∀ (n : ℕ), 0 < n → Function.Surjective (f₀ n) := by
+  intros n hn₀
+  refine Continuous.surjective (hmo₄ n hn₀) ?_ ?_
+  . refine Monotone.tendsto_atTop_atTop ?_ ?_
+    . exact StrictMono.monotone (hmo₂ n hn₀)
+    . intro b
+      use (b + 1)
+      refine Nat.le_induction ?_ ?_ n hn₀
+      . rw [hf₂ 1 (b + 1) (by linarith), h₀]
+        simp
+      . intros d hd₀ hd₁
+        rw [hf₂ (d + 1) (b + 1) (by linarith), h₁ d (b + 1) (by linarith)]
+        have hd₂: b ≤ f d (b + 1) := by
+          rw [hf₂ d (b + 1) (by linarith)] at hd₁
+          exact (Real.le_toNNReal_iff_coe_le (h₃ d (b + 1) hd₀)).mp hd₁
+        have hd₃: 1 < (f d (b + 1) + 1 / ↑d) := by
+          by_cases hd₄: 1 < d
+          . refine lt_add_of_lt_of_pos ?_ ?_
+            . refine h₅ d (b + 1) ?_
+              constructor
+              . exact hd₄
+              . exact le_add_self
+            . refine div_pos (by linarith) ?_
+              exact Nat.cast_pos'.mpr hd₀
+          . have hd₅: d = 1 := by linarith
+            rw [hd₅, h₀]
+            simp
+            norm_cast
+            refine add_pos_of_nonneg_of_pos ?_ ?_
+            . exact _root_.zero_le b
+            . exact zero_lt_one' NNReal
+        refine NNReal.le_toNNReal_of_coe_le ?_
+        nth_rw 1 [← mul_one (↑b:ℝ)]
+        refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+        exact h₃ d (b + 1) hd₀
+  . refine Filter.tendsto_atBot_atBot.mpr ?_
+    intro b
+    use 0
+    intro a ha₀
+    have ha₁: a = 0 := by exact nonpos_iff_eq_zero.mp ha₀
+    have ha₂: f₀ n 0 = 0 := by
+      refine Nat.le_induction ?_ ?_ n hn₀
+      . rw [hf₂ 1 0 (by linarith), h₀]
+        exact Real.toNNReal_coe
+      . intros d hd₀ hd₁
+        rw [hf₂ (d + 1) 0 (by linarith), h₁ d 0 (by linarith)]
+        have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+        have hd₃: f d 0 = 0 := by
+          rw [hf₂ d 0 (by linarith)] at hd₁
+          apply Real.toNNReal_eq_zero.mp at hd₁
+          exact eq_of_le_of_le hd₁ hd₂
+        rw [hd₃, zero_mul]
+        exact Real.toNNReal_zero
+    rw [ha₁, ha₂]
+    exact _root_.zero_le b
+
+
+
+lemma imo_1985_p6_7_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Filter.Tendsto (f₀ n) Filter.atTop Filter.atTop := by
+  refine Monotone.tendsto_atTop_atTop ?_ ?_
+  . exact StrictMono.monotone (hmo₂ n hn₀)
+  . intro b
+    use (b + 1)
+    refine Nat.le_induction ?_ ?_ n hn₀
+    . rw [hf₂ 1 (b + 1) (by linarith), h₀]
+      simp
+    . intros d hd₀ hd₁
+      rw [hf₂ (d + 1) (b + 1) (by linarith), h₁ d (b + 1) (by linarith)]
+      have hd₂: b ≤ f d (b + 1) := by
+        rw [hf₂ d (b + 1) (by linarith)] at hd₁
+        exact (Real.le_toNNReal_iff_coe_le (h₃ d (b + 1) hd₀)).mp hd₁
+      have hd₃: 1 < (f d (b + 1) + 1 / ↑d) := by
+        by_cases hd₄: 1 < d
+        . refine lt_add_of_lt_of_pos ?_ ?_
+          . refine h₅ d (b + 1) ?_
+            constructor
+            . exact hd₄
+            . exact le_add_self
+          . refine div_pos (by linarith) ?_
+            exact Nat.cast_pos'.mpr hd₀
+        . have hd₅: d = 1 := by linarith
+          rw [hd₅, h₀]
+          simp
+          norm_cast
+          refine add_pos_of_nonneg_of_pos ?_ ?_
+          . exact _root_.zero_le b
+          . exact zero_lt_one' NNReal
+      refine NNReal.le_toNNReal_of_coe_le ?_
+      nth_rw 1 [← mul_one (↑b:ℝ)]
+      refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+      exact h₃ d (b + 1) hd₀
+
+
+lemma imo_1985_p6_7_2
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (b : NNReal):
+   ∃ a, b ≤ f₀ n a := by
+  use (b + 1)
+  refine Nat.le_induction ?_ ?_ n hn₀
+  . rw [hf₂ 1 (b + 1) (by linarith), h₀]
+    simp
+  . intros d hd₀ hd₁
+    rw [hf₂ (d + 1) (b + 1) (by linarith), h₁ d (b + 1) (by linarith)]
+    have hd₂: b ≤ f d (b + 1) := by
+      rw [hf₂ d (b + 1) (by linarith)] at hd₁
+      exact (Real.le_toNNReal_iff_coe_le (h₃ d (b + 1) hd₀)).mp hd₁
+    have hd₃: 1 < (f d (b + 1) + 1 / ↑d) := by
+      by_cases hd₄: 1 < d
+      . refine lt_add_of_lt_of_pos ?_ ?_
+        . refine h₅ d (b + 1) ?_
+          constructor
+          . exact hd₄
+          . exact le_add_self
+        . refine div_pos (by linarith) ?_
+          exact Nat.cast_pos'.mpr hd₀
+      . have hd₅: d = 1 := by linarith
+        rw [hd₅, h₀]
+        simp
+        norm_cast
+        refine add_pos_of_nonneg_of_pos ?_ ?_
+        . exact _root_.zero_le b
+        . exact zero_lt_one' NNReal
+    refine NNReal.le_toNNReal_of_coe_le ?_
+    nth_rw 1 [← mul_one (↑b:ℝ)]
+    refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+    exact h₃ d (b + 1) hd₀
+
+
+lemma imo_1985_p6_7_3
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (b : NNReal):
+   b ≤ f₀ n (b + 1) := by
+  refine Nat.le_induction ?_ ?_ n hn₀
+  . rw [hf₂ 1 (b + 1) (by linarith), h₀]
+    simp
+  . intros d hd₀ hd₁
+    rw [hf₂ (d + 1) (b + 1) (by linarith), h₁ d (b + 1) (by linarith)]
+    have hd₂: b ≤ f d (b + 1) := by
+      rw [hf₂ d (b + 1) (by linarith)] at hd₁
+      exact (Real.le_toNNReal_iff_coe_le (h₃ d (b + 1) hd₀)).mp hd₁
+    have hd₃: 1 < (f d (b + 1) + 1 / ↑d) := by
+      by_cases hd₄: 1 < d
+      . refine lt_add_of_lt_of_pos ?_ ?_
+        . refine h₅ d (b + 1) ?_
+          constructor
+          . exact hd₄
+          . exact le_add_self
+        . refine div_pos (by linarith) ?_
+          exact Nat.cast_pos'.mpr hd₀
+      . have hd₅: d = 1 := by linarith
+        rw [hd₅, h₀]
+        simp
+        norm_cast
+        refine add_pos_of_nonneg_of_pos ?_ ?_
+        . exact _root_.zero_le b
+        . exact zero_lt_one' NNReal
+    refine NNReal.le_toNNReal_of_coe_le ?_
+    nth_rw 1 [← mul_one (↑b:ℝ)]
+    refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+    exact h₃ d (b + 1) hd₀
+
+lemma imo_1985_p6_7_4
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (b : NNReal):
+   b ≤ f₀ (Nat.succ 0) (b + 1) := by
+  rw [hf₂ 1 (b + 1) (by linarith), h₀]
+  simp
+
+
+lemma imo_1985_p6_7_5
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (b : NNReal):
+   ∀ (n : ℕ), Nat.succ 0 ≤ n → b ≤ f₀ n (b + 1) → b ≤ f₀ (n + 1) (b + 1) := by
+  intros d hd₀ hd₁
+  rw [hf₂ (d + 1) (b + 1) (by linarith), h₁ d (b + 1) (by linarith)]
+  have hd₂: b ≤ f d (b + 1) := by
+    rw [hf₂ d (b + 1) (by linarith)] at hd₁
+    exact (Real.le_toNNReal_iff_coe_le (h₃ d (b + 1) hd₀)).mp hd₁
+  have hd₃: 1 < (f d (b + 1) + 1 / ↑d) := by
+    by_cases hd₄: 1 < d
+    . refine lt_add_of_lt_of_pos ?_ ?_
+      . refine h₅ d (b + 1) ?_
+        constructor
+        . exact hd₄
+        . exact le_add_self
+      . refine div_pos (by linarith) ?_
+        exact Nat.cast_pos'.mpr hd₀
+    . have hd₅: d = 1 := by linarith
+      rw [hd₅, h₀]
+      simp
+      norm_cast
+      refine add_pos_of_nonneg_of_pos ?_ ?_
+      . exact _root_.zero_le b
+      . exact zero_lt_one' NNReal
+  refine NNReal.le_toNNReal_of_coe_le ?_
+  nth_rw 1 [← mul_one (↑b:ℝ)]
+  refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+  exact h₃ d (b + 1) hd₀
+
+
+lemma imo_1985_p6_7_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₁ : b ≤ f₀ d (b + 1)):
+   b ≤ (f d (b + 1) * (f d (b + 1) + 1 / ↑d)).toNNReal := by
+  have hd₂: b ≤ f d (b + 1) := by
+    rw [hf₂ d (b + 1) (by linarith)] at hd₁
+    exact (Real.le_toNNReal_iff_coe_le (h₃ d (b + 1) hd₀)).mp hd₁
+  have hd₃: 1 < (f d (b + 1) + 1 / ↑d) := by
+    by_cases hd₄: 1 < d
+    . refine lt_add_of_lt_of_pos ?_ ?_
+      . refine h₅ d (b + 1) ?_
+        constructor
+        . exact hd₄
+        . exact le_add_self
+      . refine div_pos (by linarith) ?_
+        exact Nat.cast_pos'.mpr hd₀
+    . have hd₅: d = 1 := by linarith
+      rw [hd₅, h₀]
+      simp
+      norm_cast
+      refine add_pos_of_nonneg_of_pos ?_ ?_
+      . exact _root_.zero_le b
+      . exact zero_lt_one' NNReal
+  refine NNReal.le_toNNReal_of_coe_le ?_
+  nth_rw 1 [← mul_one (↑b:ℝ)]
+  refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+  exact h₃ d (b + 1) hd₀
+
+lemma imo_1985_p6_7_7
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₁ : b ≤ f₀ d (b + 1)):
+   ↑b ≤ f d (b + 1) := by
+  rw [hf₂ d (b + 1) (by linarith)] at hd₁
+  exact (Real.le_toNNReal_iff_coe_le (h₃ d (b + 1) hd₀)).mp hd₁
+
+
+lemma imo_1985_p6_7_8
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₂ : ↑b ≤ f d (b + 1)):
+   b ≤ (f d (b + 1) * (f d (b + 1) + 1 / ↑d)).toNNReal := by
+  have hd₃: 1 < (f d (b + 1) + 1 / ↑d) := by
+    by_cases hd₄: 1 < d
+    . refine lt_add_of_lt_of_pos ?_ ?_
+      . refine h₅ d (b + 1) ?_
+        constructor
+        . exact hd₄
+        . exact le_add_self
+      . refine div_pos (by linarith) ?_
+        exact Nat.cast_pos'.mpr hd₀
+    . have hd₅: d = 1 := by linarith
+      rw [hd₅, h₀]
+      simp
+      norm_cast
+      refine add_pos_of_nonneg_of_pos ?_ ?_
+      . exact _root_.zero_le b
+      . exact zero_lt_one' NNReal
+  refine NNReal.le_toNNReal_of_coe_le ?_
+  nth_rw 1 [← mul_one (↑b:ℝ)]
+  refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+  exact h₃ d (b + 1) hd₀
+
+lemma imo_1985_p6_7_9
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d):
+   1 < f d (b + 1) + 1 / ↑d := by
+  by_cases hd₄: 1 < d
+  . refine lt_add_of_lt_of_pos ?_ ?_
+    . refine h₅ d (b + 1) ?_
+      constructor
+      . exact hd₄
+      . exact le_add_self
+    . refine div_pos (by linarith) ?_
+      exact Nat.cast_pos'.mpr hd₀
+  . have hd₅: d = 1 := by linarith
+    rw [hd₅, h₀]
+    simp
+    norm_cast
+    refine add_pos_of_nonneg_of_pos ?_ ?_
+    . exact _root_.zero_le b
+    . exact zero_lt_one' NNReal
+
+
+lemma imo_1985_p6_7_10
+  (f : ℕ → NNReal → ℝ)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₄ : 1 < d):
+   1 < f d (b + 1) + 1 / ↑d := by
+  refine lt_add_of_lt_of_pos ?_ ?_
+  . refine h₅ d (b + 1) ?_
+    constructor
+    . exact hd₄
+    . exact le_add_self
+  . refine div_pos (by linarith) ?_
+    exact Nat.cast_pos'.mpr hd₀
+
+
+lemma imo_1985_p6_7_11
+  (f : ℕ → NNReal → ℝ)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₄ : 1 < d):
+   1 < f d (b + 1) := by
+  refine h₅ d (b + 1) ?_
+  constructor
+  . exact hd₄
+  . exact le_add_self
+
+
+lemma imo_1985_p6_7_12
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d) :
+   0 < (1:ℝ) / ↑d := by
+  refine div_pos (by linarith) ?_
+  exact Nat.cast_pos'.mpr hd₀
+
+
+lemma imo_1985_p6_7_13
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₄ : ¬1 < d):
+   1 < f d (b + 1) + 1 / ↑d := by
+  have hd₅: d = 1 := by linarith
+  rw [hd₅, h₀]
+  simp
+  norm_cast
+  refine add_pos_of_nonneg_of_pos ?_ ?_
+  . exact _root_.zero_le b
+  . exact zero_lt_one' NNReal
+
+
+lemma imo_1985_p6_7_14
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₄ : ¬1 < d)
+  (hd₅ : 1 < f d (b + 1) + 1 / ↑d):
+   0 < b + 1 := by
+  have hd₆: d = 1 := by linarith
+  rw [hd₆, h₀] at hd₅
+  simp at hd₅
+  norm_cast at hd₅
+
+
+lemma imo_1985_p6_7_15
+  (b : NNReal):
+   0 < b + 1 := by
+  refine add_pos_of_nonneg_of_pos ?_ ?_
+  . exact _root_.zero_le b
+  . exact zero_lt_one' NNReal
+
+
+
+lemma imo_1985_p6_7_16
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₂ : ↑b ≤ f d (b + 1))
+  (hd₃ : 1 < f d (b + 1) + 1 / ↑d):
+   b ≤ (f d (b + 1) * (f d (b + 1) + 1 / ↑d)).toNNReal := by
+  refine NNReal.le_toNNReal_of_coe_le ?_
+  nth_rw 1 [← mul_one (↑b:ℝ)]
+  refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+  exact h₃ d (b + 1) hd₀
+
+
+lemma imo_1985_p6_7_17
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₂ : ↑b ≤ f d (b + 1))
+  (hd₃ : 1 < f d (b + 1) + 1 / ↑d):
+   ↑b ≤ f d (b + 1) * (f d (b + 1) + 1 / ↑d) := by
+  nth_rw 1 [← mul_one (↑b:ℝ)]
+  refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+  exact h₃ d (b + 1) hd₀
+
+
+lemma imo_1985_p6_7_18
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₂ : ↑b ≤ f d (b + 1))
+  (hd₃ : 1 < f d (b + 1) + 1 / ↑d):
+   ↑b * 1 ≤ f d (b + 1) * (f d (b + 1) + 1 / ↑d) := by
+  refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+  exact h₃ d (b + 1) hd₀
+
+
+lemma imo_1985_p6_7_19
+  (f : ℕ → NNReal → ℝ)
+  (b : NNReal)
+  (d : ℕ)
+  (hd₄ : ↑b * 1 ≤ f d (b + 1) * (f d (b + 1) + 1 / ↑d)):
+   b ≤ (f d (b + 1) * (f d (b + 1) + 1 / ↑d)).toNNReal := by
+  refine NNReal.le_toNNReal_of_coe_le ?_
+  nth_rw 1 [← mul_one (↑b:ℝ)]
+  exact hd₄
+
+
+lemma imo_1985_p6_7_20
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Filter.Tendsto (f₀ n) Filter.atBot Filter.atBot := by
+  refine Filter.tendsto_atBot_atBot.mpr ?_
+  intro b
+  use 0
+  intro a ha₀
+  have ha₁: a = 0 := by exact nonpos_iff_eq_zero.mp ha₀
+  have ha₂: f₀ n 0 = 0 := by
+    refine Nat.le_induction ?_ ?_ n hn₀
+    . rw [hf₂ 1 0 (by linarith), h₀]
+      exact Real.toNNReal_coe
+    . intros d hd₀ hd₁
+      rw [hf₂ (d + 1) 0 (by linarith), h₁ d 0 (by linarith)]
+      have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+      have hd₃: f d 0 = 0 := by
+        rw [hf₂ d 0 (by linarith)] at hd₁
+        apply Real.toNNReal_eq_zero.mp at hd₁
+        exact eq_of_le_of_le hd₁ hd₂
+      rw [hd₃, zero_mul]
+      exact Real.toNNReal_zero
+  rw [ha₁, ha₂]
+  exact _root_.zero_le b
+
+
+lemma imo_1985_p6_7_21
+  (f₀ : ℕ → NNReal → NNReal)
+  (n : ℕ)
+  (b a : NNReal)
+  (ha₁ : a = 0)
+  (ha₂ : f₀ n 0 = 0):
+   f₀ n a ≤ b := by
+  rw [ha₁, ha₂]
+  exact _root_.zero_le b
+
+lemma imo_1985_p6_7_22
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   ∀ (b : NNReal), ∃ i, ∀ a ≤ i, f₀ n a ≤ b := by
+  intro b
+  use 0
+  intro a ha₀
+  have ha₁: a = 0 := by exact nonpos_iff_eq_zero.mp ha₀
+  have ha₂: f₀ n 0 = 0 := by
+    refine Nat.le_induction ?_ ?_ n hn₀
+    . rw [hf₂ 1 0 (by linarith), h₀]
+      exact Real.toNNReal_coe
+    . intros d hd₀ hd₁
+      rw [hf₂ (d + 1) 0 (by linarith), h₁ d 0 (by linarith)]
+      have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+      have hd₃: f d 0 = 0 := by
+        rw [hf₂ d 0 (by linarith)] at hd₁
+        apply Real.toNNReal_eq_zero.mp at hd₁
+        exact eq_of_le_of_le hd₁ hd₂
+      rw [hd₃, zero_mul]
+      exact Real.toNNReal_zero
+  rw [ha₁, ha₂]
+  exact _root_.zero_le b
+
+lemma imo_1985_p6_7_23
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (b : NNReal):
+   ∃ i, ∀ a ≤ i, f₀ n a ≤ b := by
+  use 0
+  intro a ha₀
+  have ha₁: a = 0 := by exact nonpos_iff_eq_zero.mp ha₀
+  have ha₂: f₀ n 0 = 0 := by
+    refine Nat.le_induction ?_ ?_ n hn₀
+    . rw [hf₂ 1 0 (by linarith), h₀]
+      exact Real.toNNReal_coe
+    . intros d hd₀ hd₁
+      rw [hf₂ (d + 1) 0 (by linarith), h₁ d 0 (by linarith)]
+      have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+      have hd₃: f d 0 = 0 := by
+        rw [hf₂ d 0 (by linarith)] at hd₁
+        apply Real.toNNReal_eq_zero.mp at hd₁
+        exact eq_of_le_of_le hd₁ hd₂
+      rw [hd₃, zero_mul]
+      exact Real.toNNReal_zero
+  rw [ha₁, ha₂]
+  exact _root_.zero_le b
+
+lemma imo_1985_p6_7_24
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (b a : NNReal)
+  (ha₀ : a ≤ 0):
+   f₀ n a ≤ b := by
+  have ha₁: a = 0 := by exact nonpos_iff_eq_zero.mp ha₀
+  have ha₂: f₀ n 0 = 0 := by
+    refine Nat.le_induction ?_ ?_ n hn₀
+    . rw [hf₂ 1 0 (by linarith), h₀]
+      exact Real.toNNReal_coe
+    . intros d hd₀ hd₁
+      rw [hf₂ (d + 1) 0 (by linarith), h₁ d 0 (by linarith)]
+      have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+      have hd₃: f d 0 = 0 := by
+        rw [hf₂ d 0 (by linarith)] at hd₁
+        apply Real.toNNReal_eq_zero.mp at hd₁
+        exact eq_of_le_of_le hd₁ hd₂
+      rw [hd₃, zero_mul]
+      exact Real.toNNReal_zero
+  rw [ha₁, ha₂]
+  exact _root_.zero_le b
+
+lemma imo_1985_p6_7_25
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (b a : NNReal)
+  (ha₁ : a = 0):
+   f₀ n a ≤ b := by
+  have ha₂: f₀ n 0 = 0 := by
+    refine Nat.le_induction ?_ ?_ n hn₀
+    . rw [hf₂ 1 0 (by linarith), h₀]
+      exact Real.toNNReal_coe
+    . intros d hd₀ hd₁
+      rw [hf₂ (d + 1) 0 (by linarith), h₁ d 0 (by linarith)]
+      have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+      have hd₃: f d 0 = 0 := by
+        rw [hf₂ d 0 (by linarith)] at hd₁
+        apply Real.toNNReal_eq_zero.mp at hd₁
+        exact eq_of_le_of_le hd₁ hd₂
+      rw [hd₃, zero_mul]
+      exact Real.toNNReal_zero
+  rw [ha₁, ha₂]
+  exact _root_.zero_le b
+
+lemma imo_1985_p6_7_26
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   f₀ n 0 = 0 := by
+  refine Nat.le_induction ?_ ?_ n hn₀
+  . rw [hf₂ 1 0 (by linarith), h₀]
+    exact Real.toNNReal_coe
+  . intros d hd₀ hd₁
+    rw [hf₂ (d + 1) 0 (by linarith), h₁ d 0 (by linarith)]
+    have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+    have hd₃: f d 0 = 0 := by
+      rw [hf₂ d 0 (by linarith)] at hd₁
+      apply Real.toNNReal_eq_zero.mp at hd₁
+      exact eq_of_le_of_le hd₁ hd₂
+    rw [hd₃, zero_mul]
+    exact Real.toNNReal_zero
+
+
+lemma imo_1985_p6_7_27
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal):
+   f₀ (Nat.succ 0) 0 = 0 := by
+  rw [hf₂ 1 0 (by linarith), h₀]
+  exact Real.toNNReal_coe
+
+
+lemma imo_1985_p6_7_28
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal):
+   ∀ (n : ℕ), Nat.succ 0 ≤ n → f₀ n 0 = 0 → f₀ (n + 1) 0 = 0 := by
+  intros d hd₀ hd₁
+  rw [hf₂ (d + 1) 0 (by linarith), h₁ d 0 (by linarith)]
+  have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+  have hd₃: f d 0 = 0 := by
+    rw [hf₂ d 0 (by linarith)] at hd₁
+    apply Real.toNNReal_eq_zero.mp at hd₁
+    exact eq_of_le_of_le hd₁ hd₂
+  rw [hd₃, zero_mul]
+  exact Real.toNNReal_zero
+
+lemma imo_1985_p6_7_29
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₁ : f₀ d 0 = 0):
+   (f d 0 * (f d 0 + 1 / ↑d)).toNNReal = 0 := by
+  have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+  have hd₃: f d 0 = 0 := by
+    rw [hf₂ d 0 (by linarith)] at hd₁
+    apply Real.toNNReal_eq_zero.mp at hd₁
+    exact eq_of_le_of_le hd₁ hd₂
+  rw [hd₃, zero_mul]
+  exact Real.toNNReal_zero
+
+lemma imo_1985_p6_7_30
+  (f : ℕ → NNReal → ℝ)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₁ : f₀ d 0 = 0)
+  (hd₂ : 0 ≤ f d 0):
+   (f d 0 * (f d 0 + 1 / ↑d)).toNNReal = 0 := by
+  have hd₃: f d 0 = 0 := by
+    rw [hf₂ d 0 (by linarith)] at hd₁
+    apply Real.toNNReal_eq_zero.mp at hd₁
+    exact eq_of_le_of_le hd₁ hd₂
+  rw [hd₃, zero_mul]
+  exact Real.toNNReal_zero
+
+
+lemma imo_1985_p6_7_31
+  (f : ℕ → NNReal → ℝ)
+  (d : ℕ)
+  (hd₃ : f d 0 = 0):
+   (f d 0 * (f d 0 + 1 / ↑d)).toNNReal = 0 := by
+  rw [hd₃, zero_mul]
+  exact Real.toNNReal_zero
+
+
+lemma imo_1985_p6_7_32
+  (f : ℕ → NNReal → ℝ)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (d : ℕ)
+  (hd₀ : Nat.succ 0 ≤ d)
+  (hd₁ : f₀ d 0 = 0)
+  (hd₂ : 0 ≤ f d 0):
+   f d 0 = 0 := by
+  rw [hf₂ d 0 (by linarith)] at hd₁
+  apply Real.toNNReal_eq_zero.mp at hd₁
+  exact eq_of_le_of_le hd₁ hd₂
+
+
+lemma imo_1985_p6_8
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hsn₁ : ∀ (n : ↑sn), ↑n ∈ sn ∧ 0 < n.1)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n) :
+  ∀ (n : ↑sn), fb n < 1 := by
+  intros n
+  have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+  let z := fb n
+  have hz₀: z = fb n := by rfl
+  rw [← hz₀]
+  by_contra! hc₀
+  have hc₁: 1 ≤ f n z := by
+    by_cases hn₁: 1 < (n:ℕ)
+    . refine le_of_lt ?_
+      refine imo_1985_p6_3 f h₀ h₁ ?_ (↑n) z ?_
+      . exact fun n x y a a_1 ↦ hmo₀ n a a_1
+      . exact ⟨hn₁, hc₀⟩
+    . have hn₂: (n:ℕ) = 1 := by linarith
+      rw [hn₂, h₀]
+      exact hc₀
+  have hz₁: f₀ n z = 1 - 1 / n := by
+    exact hfb₁ n
+  have hz₃: f n z = 1 - 1 / n := by
+    rw [hf₂ n z hn₀] at hz₁
+    by_cases hn₁: 1 < (n:ℕ)
+    . have hz₂: 1 - 1 / (n:NNReal) ≠ 0 := by
+        have g₀: (n:NNReal) ≠ 0 := by
+          norm_cast
+          linarith
+        nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+        refine div_ne_zero ?_ g₀
+        norm_cast
+        exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+      apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+      rw [hz₁]
+      exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+    . have hn₂: (n:ℕ) = 1 := by linarith
+      rw [hn₂, h₀] at hz₁
+      simp at hz₁
+      rw [hn₂, h₀, hz₁]
+      simp
+  rw [hz₃] at hc₁
+  have hz₄: 0 < 1 / (n:ℝ) := by
+    refine div_pos (by linarith) ?_
+    exact Nat.cast_pos'.mpr hn₀
+  linarith
+
+
+lemma imo_1985_p6_8_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1):
+   fb n < 1 := by
+  let z := fb n
+  have hz₀: z = fb n := by rfl
+  rw [← hz₀]
+  by_contra! hc₀
+  have hc₁: 1 ≤ f n z := by
+    by_cases hn₁: 1 < (n:ℕ)
+    . refine le_of_lt ?_
+      refine imo_1985_p6_3 f h₀ h₁ ?_ (↑n) z ?_
+      . exact fun n x y a a_1 => hmo₀ n a a_1
+      . exact ⟨hn₁, hc₀⟩
+    . push_neg at hn₁
+      have hn₂: n.1 = 1 := by linarith
+      rw [hn₂, h₀]
+      exact hc₀
+  have hz₁: f₀ n z = 1 - 1 / n := by
+    exact hfb₁ n
+  have hz₃: f n z = 1 - 1 / n := by
+    rw [hf₂ n z hn₀] at hz₁
+    by_cases hn₁: 1 < (n:ℕ)
+    . have hz₂: 1 - 1 / (n:NNReal) ≠ 0 := by
+        have g₀: (n:NNReal) ≠ 0 := by
+          norm_cast
+          linarith
+        nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+        refine div_ne_zero ?_ g₀
+        norm_cast
+        exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+      apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+      rw [hz₁]
+      exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+    . have hn₂: (n:ℕ) = 1 := by linarith
+      rw [hn₂, h₀] at hz₁
+      simp at hz₁
+      rw [hn₂, h₀, hz₁]
+      simp
+  rw [hz₃] at hc₁
+  have hz₄: 0 < 1 / (n:ℝ) := by
+    refine div_pos (by linarith) ?_
+    exact Nat.cast_pos'.mpr hn₀
+  linarith
+
+lemma imo_1985_p6_8_2
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hz₀ : z = fb n)
+  (hc₀ : 1 ≤ z):
+   False := by
+  have hc₁: 1 ≤ f n z := by
+    by_cases hn₁: 1 < (n:ℕ)
+    . refine le_of_lt ?_
+      refine imo_1985_p6_3 f h₀ h₁ ?_ (↑n) z ?_
+      . exact fun n x y a a_1 => hmo₀ n a a_1
+      . exact ⟨hn₁, hc₀⟩
+    . have hn₂: (n:ℕ) = 1 := by linarith
+      rw [hn₂, h₀]
+      exact hc₀
+  have hz₁: f₀ n z = 1 - 1 / n := by
+    rw [hz₀]
+    exact hfb₁ n
+  have hz₃: f n z = 1 - 1 / n := by
+    rw [hf₂ n z hn₀] at hz₁
+    by_cases hn₁: 1 < (n:ℕ)
+    . have hz₂: 1 - 1 / (n:NNReal) ≠ 0 := by
+        have g₀: (n:NNReal) ≠ 0 := by
+          norm_cast
+          linarith
+        nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+        refine div_ne_zero ?_ g₀
+        norm_cast
+        exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+      apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+      rw [hz₁]
+      exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+    . have hn₂: (n:ℕ) = 1 := by linarith
+      rw [hn₂, h₀] at hz₁
+      simp at hz₁
+      rw [hn₂, h₀, hz₁]
+      simp
+  rw [hz₃] at hc₁
+  have hz₄: 0 < 1 / (n:ℝ) := by
+    refine div_pos (by linarith) ?_
+    exact Nat.cast_pos'.mpr hn₀
+  linarith
+
+
+lemma imo_1985_p6_8_3
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hc₀ : 1 ≤ z):
+   1 ≤ f (↑n) z := by
+  by_cases hn₁: 1 < (n:ℕ)
+  . refine le_of_lt ?_
+    refine imo_1985_p6_3 f h₀ h₁ ?_ (↑n) z ?_
+    . exact fun n x y a a_1 => hmo₀ n a a_1
+    . exact ⟨hn₁, hc₀⟩
+  . have hn₂: (n:ℕ) = 1 := by linarith
+    rw [hn₂, h₀]
+    exact hc₀
+
+lemma imo_1985_p6_8_4
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (z : NNReal)
+  (hc₀ : 1 ≤ z)
+  (hn₁ : 1 < n.1):
+   1 ≤ f (↑n) z := by
+  refine le_of_lt ?_
+  refine imo_1985_p6_3 f h₀ h₁ ?_ (↑n) z ?_
+  . exact fun n x y a a_1 => hmo₀ n a a_1
+  . exact ⟨hn₁, hc₀⟩
+
+lemma imo_1985_p6_8_5
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (z : NNReal)
+  (hc₀ : 1 ≤ z)
+  (hn₁ : 1 < n.1):
+   1 < f (↑n) z := by
+  refine imo_1985_p6_3 f h₀ h₁ ?_ (↑n) z ?_
+  . exact fun n x y a a_1 => hmo₀ n a a_1
+  . exact ⟨hn₁, hc₀⟩
+
+lemma imo_1985_p6_8_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hc₀ : 1 ≤ z)
+  (hn₁ : ¬1 < n.1):
+   1 ≤ f (↑n) z := by
+  have hn₂: (n:ℕ) = 1 := by linarith
+  rw [hn₂, h₀]
+  exact hc₀
+
+lemma imo_1985_p6_8_7
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hz₀ : z = fb n)
+  (hc₁ : 1 ≤ f (↑n) z):
+   False := by
+  have hz₁: f₀ n z = 1 - 1 / n := by
+    rw [hz₀]
+    exact hfb₁ n
+  have hz₃: f n z = 1 - 1 / n := by
+    rw [hf₂ n z hn₀] at hz₁
+    by_cases hn₁: 1 < (n:ℕ)
+    . have hz₂: 1 - 1 / (n:NNReal) ≠ 0 := by
+        have g₀: (n:NNReal) ≠ 0 := by
+          norm_cast
+          linarith
+        nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+        refine div_ne_zero ?_ g₀
+        norm_cast
+        exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+      apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+      rw [hz₁]
+      exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+    . have hn₂: (n:ℕ) = 1 := by linarith
+      rw [hn₂, h₀] at hz₁
+      simp at hz₁
+      rw [hn₂, h₀, hz₁]
+      simp
+  rw [hz₃] at hc₁
+  have hz₄: 0 < 1 / (n:ℝ) := by
+    refine div_pos (by linarith) ?_
+    exact Nat.cast_pos'.mpr hn₀
+  linarith
+
+lemma imo_1985_p6_8_8
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hc₁ : 1 ≤ f (↑n) z)
+  (hz₁ : f₀ (↑n) z = 1 - 1 / ↑↑n):
+   False := by
+  have hz₃: f n z = 1 - 1 / n := by
+    rw [hf₂ n z hn₀] at hz₁
+    by_cases hn₁: 1 < (n:ℕ)
+    . have hz₂: 1 - 1 / (n:NNReal) ≠ 0 := by
+        have g₀: (n:NNReal) ≠ 0 := by
+          norm_cast
+          linarith
+        nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+        refine div_ne_zero ?_ g₀
+        norm_cast
+        exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+      apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+      rw [hz₁]
+      exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+    . have hn₂: (n:ℕ) = 1 := by linarith
+      rw [hn₂, h₀] at hz₁
+      simp at hz₁
+      rw [hn₂, h₀, hz₁]
+      simp
+  rw [hz₃] at hc₁
+  have hz₄: 0 < 1 / (n:ℝ) := by
+    refine div_pos (by linarith) ?_
+    exact Nat.cast_pos'.mpr hn₀
+  linarith
+
+lemma imo_1985_p6_8_9
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hz₁ : f₀ (↑n) z = 1 - 1 / ↑↑n):
+   f (↑n) z = 1 - 1 / ↑↑n := by
+  rw [hf₂ n z hn₀] at hz₁
+  by_cases hn₁: 1 < (n:ℕ)
+  . have hz₂: 1 - 1 / (n:NNReal) ≠ 0 := by
+      have g₀: (n:NNReal) ≠ 0 := by
+        norm_cast
+        linarith
+      nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+      refine div_ne_zero ?_ g₀
+      norm_cast
+      exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+    apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+    rw [hz₁]
+    exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+  . have hn₂: (n:ℕ) = 1 := by linarith
+    rw [hn₂, h₀] at hz₁
+    simp at hz₁
+    rw [hn₂, h₀, hz₁]
+    simp
+
+lemma imo_1985_p6_8_10
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hz₁ : (f (↑n) z).toNNReal = 1 - 1 / ↑↑n):
+   f (↑n) z = 1 - 1 / ↑↑n := by
+  by_cases hn₁: 1 < (n:ℕ)
+  . have hz₂: 1 - 1 / (n:NNReal) ≠ 0 := by
+      have g₀: (n:NNReal) ≠ 0 := by
+        norm_cast
+        linarith
+      nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+      refine div_ne_zero ?_ g₀
+      norm_cast
+      exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+    apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+    rw [hz₁]
+    exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+  . have hn₂: (n:ℕ) = 1 := by linarith
+    rw [hn₂, h₀] at hz₁
+    simp at hz₁
+    rw [hn₂, h₀, hz₁]
+    simp
+
+lemma imo_1985_p6_8_11
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hz₁ : (f (↑n) z).toNNReal = 1 - 1 / ↑↑n)
+  (hn₁ : 1 < n.1):
+   f (↑n) z = 1 - 1 / ↑↑n := by
+  have hz₂: 1 - 1 / (n:NNReal) ≠ 0 := by
+    have g₀: (n:NNReal) ≠ 0 := by
+      norm_cast
+      linarith
+    nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+    refine div_ne_zero ?_ g₀
+    norm_cast
+    exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+  apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+  rw [hz₁]
+  exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+
+
+lemma imo_1985_p6_8_12
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₁ : 1 < n.1):
+   1 - (1:NNReal) / n.1 ≠ 0 := by
+  have g₀: ↑(n.1) ≠ (0:NNReal) := by
+    norm_cast
+    linarith
+  nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+  refine div_ne_zero ?_ g₀
+  norm_cast
+  exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+
+
+lemma imo_1985_p6_8_13
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₁ : 1 < n.1)
+  (g₀ : ↑(n.1) ≠ (0:NNReal)):
+   1 - (1:NNReal) / ↑↑n ≠ 0 := by
+  nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+  refine div_ne_zero ?_ g₀
+  norm_cast
+  exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+
+
+lemma imo_1985_p6_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n)) :
+  ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x := by
+  intros n x hp
+  cases' hp with hn₀ hx₀
+  by_cases hn₁: 1 < n
+  . refine Nat.le_induction ?_ ?_ n hn₁
+    . rw [h₁ 1 x (by norm_num)]
+      rw [h₀ x]
+      refine mul_pos hx₀ ?_
+      refine add_pos hx₀ (by norm_num)
+    . intros m hm₀ hm₁
+      rw [h₁ m x (by linarith)]
+      refine mul_pos hm₁ ?_
+      refine add_pos hm₁ ?_
+      refine one_div_pos.mpr ?_
+      norm_cast
+      exact Nat.zero_lt_of_lt hm₀
+  . interval_cases n
+    rw [h₀ x]
+    exact hx₀
+
+
+lemma imo_1985_p6_1_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (n : ℕ)
+  (x : NNReal)
+  (hx₀ : 0 < x)
+  (hn₁ : 1 < n):
+   0 < f n x := by
+  refine Nat.le_induction ?_ ?_ n hn₁
+  . rw [h₁ 1 x (by norm_num)]
+    rw [h₀ x]
+    refine mul_pos hx₀ ?_
+    refine add_pos hx₀ (by norm_num)
+  . intros m hm₀ hm₁
+    rw [h₁ m x (by linarith)]
+    refine mul_pos hm₁ ?_
+    refine add_pos hm₁ ?_
+    refine one_div_pos.mpr ?_
+    norm_cast
+    exact Nat.zero_lt_of_lt hm₀
+
+
+
+lemma imo_1985_p6_1_2
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 0 < n)
+  (hx₀ : 0 < x)
+  (hn₁ : ¬1 < n):
+   0 < f n x := by
+  interval_cases n
+  rw [h₀ x]
+  exact hx₀
+
+
+
+lemma imo_1985_p6_1_3
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (x : NNReal)
+  (hx₀ : 0 < x):
+   0 < f (Nat.succ 1) x := by
+  rw [h₁ 1 x (by norm_num)]
+  rw [h₀ x]
+  refine mul_pos hx₀ ?_
+  refine add_pos hx₀ (by norm_num)
+
+
+lemma imo_1985_p6_1_4
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (x : NNReal)
+  (hx₀ : 0 < x):
+   0 < f 1 x * (f 1 x + 1 / ↑1) := by
+  rw [h₀ x]
+  refine mul_pos hx₀ ?_
+  refine add_pos hx₀ (by norm_num)
+
+
+lemma imo_1985_p6_1_5
+  (x : NNReal)
+  (hx₀ : 0 < x):
+   0 < ↑x * (↑x + 1 / ↑1) := by
+  refine mul_pos hx₀ ?_
+  refine add_pos hx₀ (by norm_num)
+
+lemma imo_1985_p6_1_6
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (x : NNReal):
+   ∀ (n : ℕ), Nat.succ 1 ≤ n → 0 < f n x → 0 < f (n + 1) x := by
+  intros m hm₀ hm₁
+  rw [h₁ m x (by linarith)]
+  refine mul_pos hm₁ ?_
+  refine add_pos hm₁ ?_
+  refine one_div_pos.mpr ?_
+  norm_cast
+  exact Nat.zero_lt_of_lt hm₀
+
+
+lemma imo_1985_p6_1_7
+  (f : ℕ → NNReal → ℝ)
+  (x : NNReal)
+  (m : ℕ)
+  (hm₀ : Nat.succ 1 ≤ m)
+  (hm₁ : 0 < f m x):
+   0 < f m x + 1 / ↑m := by
+  have m_nonzero : m ≠ 0 :=
+    fun h => by { rw [h] at hm₀; norm_num at hm₀ }
+  have m_pos_nat : 0 < m := Nat.pos_of_ne_zero m_nonzero
+  have m_pos : 0 < (↑m : ℝ) := Nat.cast_pos.mpr m_pos_nat
+  have one_div_pos : 0 < (1 : ℝ) / (↑m : ℝ) := div_pos zero_lt_one m_pos
+  exact add_pos hm₁ one_div_pos
+
+
+lemma imo_1985_p6_1_8
+  (f : ℕ → NNReal → ℝ)
+  (x : NNReal)
+  (m : ℕ)
+  (hm₀ : Nat.succ 1 ≤ m)
+  (hm₁ : 0 < f m x):
+   0 < f m x + 1 / ↑m := by
+  refine add_pos hm₁ ?_
+  refine one_div_pos.mpr ?_
+  norm_cast
+  exact Nat.zero_lt_of_lt hm₀
+
+
+
+lemma imo_1985_p6_2_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (n : ℕ)
+  (x y : NNReal)
+  (hxy : x < y)
+  (hn₁ : 1 < n):
+   f n x < f n y := by
+  refine Nat.le_induction ?_ ?_ n hn₁
+  . rw [h₁ 1 x (by norm_num)]
+    rw [h₁ 1 y (by norm_num)]
+    norm_num
+    refine mul_lt_mul ?_ ?_ ?_ ?_
+    . rw [h₀ x, h₀ y]
+      exact hxy
+    . refine _root_.add_le_add ?_ (by norm_num)
+      rw [h₀ x, h₀ y]
+      exact le_of_lt hxy
+    . refine add_pos_of_nonneg_of_pos ?_ (by linarith)
+      rw [h₀ x]
+      exact NNReal.zero_le_coe
+    . refine le_of_lt ?_
+      refine h₂ 1 y ?_
+      norm_num
+      exact pos_of_gt hxy
+  . intros m hm₀ hm₁
+    rw [h₁ m x (by linarith)]
+    rw [h₁ m y (by linarith)]
+    refine mul_lt_mul hm₁ ?_ ?_ ?_
+    . refine _root_.add_le_add ?_ (by norm_num)
+      refine le_of_lt hm₁
+    . refine add_pos_of_nonneg_of_pos ?_ ?_
+      . exact h₃ m x (by linarith)
+      . refine one_div_pos.mpr ?_
+        norm_cast
+        exact Nat.zero_lt_of_lt hm₀
+    . refine le_of_lt ?_
+      refine h₂ m y ?_
+      constructor
+      . exact Nat.zero_lt_of_lt hm₀
+      . exact pos_of_gt hxy
+
+
+lemma imo_1985_p6_2_2
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (n : ℕ)
+  (x y : NNReal)
+  (hn : 0 < n)
+  (hxy : x < y)
+  (hn₁ : ¬1 < n):
+   f n x < f n y := by
+  interval_cases n
+  rw [h₀ x, h₀ y]
+  exact hxy
+
+
+lemma imo_1985_p6_2_3
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (x y : NNReal)
+  (hxy : x < y):
+   f (Nat.succ 1) x < f (Nat.succ 1) y := by
+  rw [h₁ 1 x (by norm_num)]
+  rw [h₁ 1 y (by norm_num)]
+  norm_num
+  refine mul_lt_mul ?_ ?_ ?_ ?_
+  . rw [h₀ x, h₀ y]
+    exact hxy
+  . refine _root_.add_le_add ?_ (by norm_num)
+    rw [h₀ x, h₀ y]
+    exact le_of_lt hxy
+  . refine add_pos_of_nonneg_of_pos ?_ (by linarith)
+    rw [h₀ x]
+    exact NNReal.zero_le_coe
+  . refine le_of_lt ?_
+    refine h₂ 1 y ?_
+    norm_num
+    exact pos_of_gt hxy
+
+
+lemma imo_1985_p6_2_4
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (x y : NNReal)
+  (hxy : x < y):
+   ∀ (n : ℕ), Nat.succ 1 ≤ n → f n x < f n y → f (n + 1) x < f (n + 1) y := by
+  intros m hm₀ hm₁
+  rw [h₁ m x (by linarith)]
+  rw [h₁ m y (by linarith)]
+  refine mul_lt_mul hm₁ ?_ ?_ ?_
+  . refine _root_.add_le_add ?_ (by norm_num)
+    refine le_of_lt hm₁
+  . refine add_pos_of_nonneg_of_pos ?_ ?_
+    . exact h₃ m x (by linarith)
+    . refine one_div_pos.mpr ?_
+      norm_cast
+      exact Nat.zero_lt_of_lt hm₀
+  . refine le_of_lt ?_
+    refine h₂ m y ?_
+    constructor
+    . exact Nat.zero_lt_of_lt hm₀
+    . exact pos_of_gt hxy
+
+
+lemma imo_1985_p6_2_5
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (x y : NNReal)
+  (hxy : x < y):
+   f 1 x * (f 1 x + 1 / ↑1) < f 1 y * (f 1 y + 1 / ↑1) := by
+  norm_num
+  refine mul_lt_mul ?_ ?_ ?_ ?_
+  . rw [h₀ x, h₀ y]
+    exact hxy
+  . refine _root_.add_le_add ?_ (by norm_num)
+    rw [h₀ x, h₀ y]
+    exact le_of_lt hxy
+  . refine add_pos_of_nonneg_of_pos ?_ (by linarith)
+    rw [h₀ x]
+    exact NNReal.zero_le_coe
+  . refine le_of_lt ?_
+    refine h₂ 1 y ?_
+    norm_num
+    exact pos_of_gt hxy
+
+
+lemma imo_1985_p6_2_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (x y : NNReal)
+  (hxy : x < y):
+   f 1 x + 1 ≤ f 1 y + 1 := by
+  refine _root_.add_le_add ?_ (by norm_num)
+  rw [h₀ x, h₀ y]
+  exact le_of_lt hxy
+
+
+lemma imo_1985_p6_2_7
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (x : NNReal):
+   0 < f 1 x + 1 := by
+  refine add_pos_of_nonneg_of_pos ?_ (by linarith)
+  rw [h₀ x]
+  exact NNReal.zero_le_coe
+
+lemma imo_1985_p6_2_8
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (x y : NNReal)
+  (hxy : x < y):
+   0 ≤ f 1 y := by
+  refine le_of_lt ?_
+  refine h₂ 1 y ?_
+  norm_num
+  exact pos_of_gt hxy
+
+
+lemma imo_1985_p6_2_9
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (x y : NNReal)
+  (hxy : x < y)
+  (m : ℕ)
+  (hm₀ : Nat.succ 1 ≤ m)
+  (hm₁ : f m x < f m y):
+   f m x * (f m x + 1 / ↑m) < f m y * (f m y + 1 / ↑m) := by
+  refine mul_lt_mul hm₁ ?_ ?_ ?_
+  . refine _root_.add_le_add ?_ (by norm_num)
+    refine le_of_lt hm₁
+  . refine add_pos_of_nonneg_of_pos ?_ ?_
+    . exact h₃ m x (by linarith)
+    . refine one_div_pos.mpr ?_
+      norm_cast
+      exact Nat.zero_lt_of_lt hm₀
+  . refine le_of_lt ?_
+    refine h₂ m y ?_
+    constructor
+    . exact Nat.zero_lt_of_lt hm₀
+    . exact pos_of_gt hxy
+
+
+lemma imo_1985_p6_2_10
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (x : NNReal)
+  (m : ℕ)
+  (hm₀ : Nat.succ 1 ≤ m):
+   0 < f m x + 1 / ↑m := by
+  refine add_pos_of_nonneg_of_pos ?_ ?_
+  . exact h₃ m x (by linarith)
+  . refine one_div_pos.mpr ?_
+    norm_cast
+    exact Nat.zero_lt_of_lt hm₀
+
+
+lemma imo_1985_p6_2_11
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (x y : NNReal)
+  (hxy : x < y)
+  (m : ℕ)
+  (hm₀ : Nat.succ 1 ≤ m):
+   0 ≤ f m y := by
+  refine le_of_lt ?_
+  refine h₂ m y ?_
+  constructor
+  . exact Nat.zero_lt_of_lt hm₀
+  . exact pos_of_gt hxy
+
+
+lemma imo_1985_p6_3_1
+  (f : ℕ → NNReal → ℝ)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 1 < n)
+  (hx₁ : 1 ≤ x):
+   f n 1 ≤ f n x := by
+  by_cases hx₂: 1 < x
+  . refine le_of_lt ?_
+    refine h₄ n 1 x ?_ hx₂
+    exact Nat.zero_lt_of_lt hn₀
+  . push_neg at hx₂
+    have hx₃: x = 1 := by exact le_antisymm hx₂ hx₁
+    rw [hx₃]
+
+
+lemma imo_1985_p6_3_2
+  (f : ℕ → NNReal → ℝ)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 1 < n)
+  (hx₂ : 1 < x):
+   f n 1 ≤ f n x := by
+  refine le_of_lt ?_
+  refine h₄ n 1 x ?_ hx₂
+  exact Nat.zero_lt_of_lt hn₀
+
+
+lemma imo_1985_p6_3_3
+  (f : ℕ → NNReal → ℝ)
+  (n : ℕ)
+  (x : NNReal)
+  (hx₁ : 1 ≤ x)
+  (hx₂ : ¬1 < x):
+   f n 1 ≤ f n x := by
+  push_neg at hx₂
+  have hx₃: x = 1 := by exact le_antisymm hx₂ hx₁
+  rw [hx₃]
+
+
+lemma imo_1985_p6_3_4
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 1 < n)
+  (g₂₀ : f n 1 ≤ f n x):
+   1 < f n x := by
+  have g₂₁: f 1 1 < f n 1 := by
+    rw [h₀]
+    refine Nat.le_induction ?_ ?_ n hn₀
+    . rw [h₁ 1 1 (by norm_num), h₀]
+      norm_num
+    . intros m hm₀ hm₁
+      rw [h₁ m 1 (by linarith)]
+      refine one_lt_mul_of_lt_of_le hm₁ ?_
+      nth_rw 1 [← add_zero 1]
+      refine add_le_add ?_ ?_
+      . exact le_of_lt hm₁
+      . refine one_div_nonneg.mpr ?_
+        exact Nat.cast_nonneg' m
+  refine lt_of_lt_of_le ?_ g₂₀
+  exact (lt_iff_lt_of_cmp_eq_cmp (congrFun (congrArg cmp (h₀ 1)) (f n 1))).mp g₂₁
+
+
+lemma imo_1985_p6_3_5
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (n : ℕ)
+  (x : NNReal)
+  (g₂₀ : f n 1 ≤ f n x)
+  (g₂₁ : f 1 1 < f n 1):
+   1 < f n x := by
+  refine lt_of_lt_of_le ?_ g₂₀
+  exact (lt_iff_lt_of_cmp_eq_cmp (congrFun (congrArg cmp (h₀ 1)) (f n 1))).mp g₂₁
+
+
+
+lemma imo_1985_p6_3_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (n : ℕ)
+  (hn₀ : 1 < n):
+   f 1 1 < f n 1 := by
+  rw [h₀]
+  refine Nat.le_induction ?_ ?_ n hn₀
+  . rw [h₁ 1 1 (by norm_num), h₀]
+    norm_num
+  . intros m hm₀ hm₁
+    rw [h₁ m 1 (by linarith)]
+    refine one_lt_mul_of_lt_of_le hm₁ ?_
+    nth_rw 1 [← add_zero 1]
+    refine add_le_add ?_ ?_
+    . exact le_of_lt hm₁
+    . refine one_div_nonneg.mpr ?_
+      exact Nat.cast_nonneg' m
+
+
+lemma imo_1985_p6_3_7
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n)):
+   ∀ (n : ℕ), Nat.succ 1 ≤ n → ↑1 < f n 1 → ↑1 < f (n + 1) 1 := by
+  intros m hm₀ hm₁
+  rw [h₁ m 1 (by linarith)]
+  refine one_lt_mul_of_lt_of_le hm₁ ?_
+  nth_rw 1 [← add_zero 1]
+  refine add_le_add ?_ ?_
+  . exact le_of_lt hm₁
+  . refine one_div_nonneg.mpr ?_
+    exact Nat.cast_nonneg' m
+
+
+lemma imo_1985_p6_3_8
+  (f : ℕ → NNReal → ℝ)
+  (m : ℕ)
+  (hm₁ : ↑1 < f m 1):
+   ↑1 < f m 1 * (f m 1 + 1 / ↑m) := by
+  refine one_lt_mul_of_lt_of_le hm₁ ?_
+  nth_rw 1 [← add_zero 1]
+  refine add_le_add ?_ ?_
+  . exact le_of_lt hm₁
+  . refine one_div_nonneg.mpr ?_
+    exact Nat.cast_nonneg' m
+
+
+lemma imo_1985_p6_3_9
+  (f : ℕ → NNReal → ℝ)
+  (m : ℕ)
+  (hm₁ : ↑1 < f m 1):
+   1 ≤ f m 1 + 1 / ↑m := by
+  nth_rw 1 [← add_zero 1]
+  refine add_le_add ?_ ?_
+  . exact le_of_lt hm₁
+  . refine one_div_nonneg.mpr ?_
+    exact Nat.cast_nonneg' m
+
+
+lemma imo_1985_p6_4_1
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Monotone (f₀ n) := by
+  refine monotone_iff_forall_lt.mpr ?_
+  intros a b hab
+  refine le_of_lt ?_
+  rw [hf₀]
+  exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n a hn₀)).mpr (h₄ n a b hn₀ hab)
+
+lemma imo_1985_p6_4_2
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   ∀ ⦃a b : NNReal⦄, a < b → f₀ n a ≤ f₀ n b := by
+  intros a b hab
+  refine le_of_lt ?_
+  rw [hf₀]
+  exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n a hn₀)).mpr (h₄ n a b hn₀ hab)
+
+
+lemma imo_1985_p6_4_3
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (a b : NNReal)
+  (hab : a < b):
+   f₀ n a < f₀ n b := by
+  rw [hf₀]
+  exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n a hn₀)).mpr (h₄ n a b hn₀ hab)
+
+lemma imo_1985_p6_4_4
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Function.Injective (f₀ n) := by
+  intros p q hpq
+  contrapose! hpq
+  apply lt_or_gt_of_ne at hpq
+  cases' hpq with hpq hpq
+  . refine ne_of_lt ?_
+    rw [hf₀]
+    exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n p hn₀)).mpr (h₄ n p q hn₀ hpq)
+  . symm
+    refine ne_of_lt ?_
+    rw [hf₀]
+    exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n q hn₀)).mpr (h₄ n q p hn₀ hpq)
+
+
+
+lemma imo_1985_p6_4_5
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (p q : NNReal)
+  (hpq : p ≠ q):
+   f₀ n p ≠ f₀ n q := by
+  apply lt_or_gt_of_ne at hpq
+  cases' hpq with hpq hpq
+  . refine ne_of_lt ?_
+    rw [hf₀]
+    exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n p hn₀)).mpr (h₄ n p q hn₀ hpq)
+  . symm
+    refine ne_of_lt ?_
+    rw [hf₀]
+    exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n q hn₀)).mpr (h₄ n q p hn₀ hpq)
+
+lemma imo_1985_p6_4_6
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (p q : NNReal)
+  (hpq : p < q):
+   f₀ n p ≠ f₀ n q := by
+  refine ne_of_lt ?_
+  rw [hf₀]
+  exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n p hn₀)).mpr (h₄ n p q hn₀ hpq)
+
+
+lemma imo_1985_p6_4_7
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (p q : NNReal)
+  (hpq : p > q):
+   f₀ n p ≠ f₀ n q := by
+  symm
+  refine ne_of_lt ?_
+  rw [hf₀]
+  exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n q hn₀)).mpr (h₄ n q p hn₀ hpq)
+
+
+lemma imo_1985_p6_5_1
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (n : ℕ)
+  (x y : NNReal)
+  (hn₀ : 0 < n)
+  (hn₁ : f₀ n x = y):
+   fi n y = x := by
+  have hf₃: ∀ n y, fi n y = Function.invFun (f₀ n) y := by
+    exact fun n y => congrFun (congrFun hfi n) y
+  rw [← hn₁, hf₃]
+  have hmo₃: ∀ n, 0 < n → Function.Injective (f₀ n) := by
+    exact fun n a => StrictMono.injective (hmo₂ n a)
+  have hn₂: (Function.invFun (f₀ n)) ∘ (f₀ n) = id := by exact Function.invFun_comp (hmo₃ n hn₀)
+  rw [Function.comp_def (Function.invFun (f₀ n)) (f₀ n)] at hn₂
+  have hn₃: (fun x => Function.invFun (f₀ n) (f₀ n x)) x = id x := by exact Eq.symm (NNReal.eq (congrArg NNReal.toReal (congrFun (id (Eq.symm hn₂)) x)))
+  exact hmo₁ n hn₀ (congrArg (f n) hn₃)
+
+lemma imo_1985_p6_5_2
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (n : ℕ)
+  (x y : NNReal)
+  (hn₀ : 0 < n)
+  (hn₁ : f₀ n x = y)
+  (hf₃ : ∀ (n : ℕ) (y : NNReal), fi n y = Function.invFun (f₀ n) y):
+   fi n y = x := by
+  rw [← hn₁, hf₃]
+  have hmo₃: ∀ n, 0 < n → Function.Injective (f₀ n) := by
+    exact fun n a => StrictMono.injective (hmo₂ n a)
+  have hn₂: (Function.invFun (f₀ n)) ∘ (f₀ n) = id := by exact Function.invFun_comp (hmo₃ n hn₀)
+  rw [Function.comp_def (Function.invFun (f₀ n)) (f₀ n)] at hn₂
+  have hn₃: (fun x => Function.invFun (f₀ n) (f₀ n x)) x = id x := by exact Eq.symm (NNReal.eq (congrArg NNReal.toReal (congrFun (id (Eq.symm hn₂)) x)))
+  exact hmo₁ n hn₀ (congrArg (f n) hn₃)
+
+lemma imo_1985_p6_5_3
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 0 < n):
+   Function.invFun (f₀ n) (f₀ n x) = x := by
+  have hmo₃: ∀ n, 0 < n → Function.Injective (f₀ n) := by
+    exact fun n a => StrictMono.injective (hmo₂ n a)
+  have hn₂: (Function.invFun (f₀ n)) ∘ (f₀ n) = id := by exact Function.invFun_comp (hmo₃ n hn₀)
+  rw [Function.comp_def (Function.invFun (f₀ n)) (f₀ n)] at hn₂
+  have hn₃: (fun x => Function.invFun (f₀ n) (f₀ n x)) x = id x := by exact Eq.symm (NNReal.eq (congrArg NNReal.toReal (congrFun (id (Eq.symm hn₂)) x)))
+  exact hmo₁ n hn₀ (congrArg (f n) hn₃)
+
+lemma imo_1985_p6_5_4
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 0 < n)
+  (hmo₃ : ∀ (n : ℕ), 0 < n → Function.Injective (f₀ n)):
+   Function.invFun (f₀ n) (f₀ n x) = x := by
+  have hn₂: (Function.invFun (f₀ n)) ∘ (f₀ n) = id := by exact Function.invFun_comp (hmo₃ n hn₀)
+  rw [Function.comp_def (Function.invFun (f₀ n)) (f₀ n)] at hn₂
+  have hn₃: (fun x => Function.invFun (f₀ n) (f₀ n x)) x = id x := by exact Eq.symm (NNReal.eq (congrArg NNReal.toReal (congrFun (id (Eq.symm hn₂)) x)))
+  exact hmo₁ n hn₀ (congrArg (f n) hn₃)
+
+
+lemma imo_1985_p6_5_5
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 0 < n)
+  (hn₂ : Function.invFun (f₀ n) ∘ f₀ n = id):
+   Function.invFun (f₀ n) (f₀ n x) = x := by
+  rw [Function.comp_def (Function.invFun (f₀ n)) (f₀ n)] at hn₂
+  have hn₃: (fun x => Function.invFun (f₀ n) (f₀ n x)) x = id x := by exact Eq.symm (NNReal.eq (congrArg NNReal.toReal (congrFun (id (Eq.symm hn₂)) x)))
+  exact hmo₁ n hn₀ (congrArg (f n) hn₃)
+
+lemma imo_1985_p6_5_6
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 0 < n)
+  (hn₂ : (fun x ↦ Function.invFun (f₀ n) (f₀ n x)) = id):
+   Function.invFun (f₀ n) (f₀ n x) = x := by
+  have hn₃: (fun x => Function.invFun (f₀ n) (f₀ n x)) x = id x := by exact Eq.symm (NNReal.eq (congrArg NNReal.toReal (congrFun (id (Eq.symm hn₂)) x)))
+  exact hmo₁ n hn₀ (congrArg (f n) hn₃)
+
+
+lemma imo_1985_p6_6_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Continuous ((fun n x ↦ (f n x).toNNReal) n) := by
+  refine Continuous.comp' ?_ ?_
+  . exact continuous_real_toNNReal
+  . refine Nat.le_induction ?_ ?_ n hn₀
+    . have hn₁: f 1 = fun (x:NNReal) => (x:ℝ) := by exact (Set.eqOn_univ (f 1) fun x => ↑x).mp fun ⦃x⦄ _ => h₀ x
+      rw [hn₁]
+      exact NNReal.continuous_coe
+    . intros d hd₀ hd₁
+      have hd₂: f (d + 1) = fun x => f d x * (f d x + 1 / ↑d) := by
+        exact (Set.eqOn_univ (f (d + 1)) fun x => f d x * (f d x + 1 / ↑d)).mp fun ⦃x⦄ _ => h₁ d x hd₀
+      rw [hd₂]
+      refine Continuous.mul hd₁ ?_
+      refine Continuous.add hd₁ ?_
+      exact continuous_const
+
+
+lemma imo_1985_p6_6_2
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Continuous (f n) := by
+  refine Nat.le_induction ?_ ?_ n hn₀
+  . have hn₁: f 1 = fun (x:NNReal) => (x:ℝ) := by exact (Set.eqOn_univ (f 1) fun x => ↑x).mp fun ⦃x⦄ _ => h₀ x
+    rw [hn₁]
+    exact NNReal.continuous_coe
+  . intros d hd₀ hd₁
+    have hd₂: f (d + 1) = fun x => f d x * (f d x + 1 / ↑d) := by
+      exact (Set.eqOn_univ (f (d + 1)) fun x => f d x * (f d x + 1 / ↑d)).mp fun ⦃x⦄ _ => h₁ d x hd₀
+    rw [hd₂]
+    refine Continuous.mul hd₁ ?_
+    refine Continuous.add hd₁ ?_
+    exact continuous_const
+
+
+lemma imo_1985_p6_6_3
+  (f : ℕ → NNReal → ℝ)
+  (f₀ : ℕ → NNReal → NNReal := fun n x => (f n x).toNNReal)
+  (hf₀ : f₀ = fun n x => (f n x).toNNReal)
+  (hmo₄:  ∀ (n : ℕ), 0 < n → Continuous (f n)):
+   ∀ (n : ℕ), 0 < n → Continuous (f₀ n) := by
+  intros n hn₀
+  rw [hf₀]
+  refine Continuous.comp' ?_ ?_
+  . exact continuous_real_toNNReal
+  . exact hmo₄ n hn₀
+
+
+lemma imo_1985_p6_bonus_1
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n => Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n)) :
+  ∀ (n : ℕ), 0 < n → Function.Bijective (f₀ n) := by
+  intros n hn₀
+  refine Function.bijective_iff_has_inverse.mpr ?_
+  use fi n
+  constructor
+  . rw [hfi]
+    refine Function.leftInverse_invFun ?_
+    exact StrictMono.injective (hmo₂ n hn₀)
+  . exact hmo₇ n hn₀
+
+
+lemma imo_1985_p6_bonus_1_1
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   ∃ g, Function.LeftInverse g (f₀ n) ∧ Function.RightInverse g (f₀ n) := by
+  use fi n
+  constructor
+  . rw [hfi]
+    refine Function.leftInverse_invFun ?_
+    exact StrictMono.injective (hmo₂ n hn₀)
+  . exact hmo₇ n hn₀
+
+
+lemma imo_1985_p6_bonus_1_2
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Function.LeftInverse (fi n) (f₀ n) ∧ Function.RightInverse (fi n) (f₀ n) := by
+  constructor
+  . rw [hfi]
+    refine Function.leftInverse_invFun ?_
+    exact StrictMono.injective (hmo₂ n hn₀)
+  . exact hmo₇ n hn₀
+
+
+lemma imo_1985_p6_bonus_1_3
+  (f₀ : ℕ → NNReal → NNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (h₁ : ∀ n, Function.LeftInverse (fi n) (f₀ n) ∧ Function.RightInverse (fi n) (f₀ n)):
+   ∀ (n : ℕ), 0 < n → Function.Bijective (f₀ n) := by
+  intros n _
+  refine Function.bijective_iff_has_inverse.mpr ?_
+  use fi n
+  exact h₁ n
+
+
+
+
+lemma imo_1985_p6_bonus_2
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n => Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n)) :
+  ∀ (n : ℕ), 0 < n → ∃! c, f₀ n c = 1 := by
+  intros n hn₀
+  refine Function.Bijective.existsUnique ?_ 1
+  refine Function.bijective_iff_has_inverse.mpr ?_
+  use fi n
+  constructor
+  . rw [hfi]
+    refine Function.leftInverse_invFun ?_
+    exact StrictMono.injective (hmo₂ n hn₀)
+  . exact hmo₇ n hn₀
+
+
+lemma imo_1985_p6_bonus_3
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (sb  : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br) :
+  br ∉ sbr := by
+  rw [hsbr]
+  by_contra! hc₀
+  apply (Set.mem_image fr sb br).mp at hc₀
+  obtain ⟨x, hx₀, hx₁⟩ := hc₀
+  rw [hsb₀] at hx₀
+  apply Set.mem_range.mp at hx₀
+  obtain ⟨nx, hnx₀⟩ := hx₀
+  have hnx₁: (nx.1 + (1:ℕ)) ∈ sn := by
+    let t : ℕ := nx.1 + 1
+    have ht₀: t = nx.1 + 1 := by rfl
+    rw [← ht₀, hsn]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_left 1 ↑nx
+  let ny : ↑sn := ⟨(nx.1 + 1), hnx₁⟩
+  have hx₂: fb nx < fb ny := by
+    refine hfb₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fb ny ∈ sb := by
+    rw [hsb₀]
+    exact Set.mem_range_self ny
+  have hx₄: fb ny ≤ br := by
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    rw [hfr]
+    use (fb ny)
+  have hc₁: br < fb ny := by
+    rw [ ← hx₁, ��� hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false br).mp ?_
+  exact lt_of_lt_of_le hc₁ hx₄
+
+
+
+
+lemma imo_1985_p6_bonus_3_1
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hc₀ : br ∈ fr '' sb):
+   False := by
+  apply (Set.mem_image fr sb br).mp at hc₀
+  obtain ⟨x, hx₀, hx₁⟩ := hc₀
+  rw [hsb₀] at hx₀
+  apply Set.mem_range.mp at hx₀
+  obtain ⟨nx, hnx₀⟩ := hx₀
+  have hnx₁: (nx.1 + (1:ℕ)) ∈ sn := by
+    let t : ℕ := nx.1 + 1
+    have ht₀: t = nx.1 + 1 := by rfl
+    rw [← ht₀, hsn]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_left 1 ↑nx
+  let ny : ↑sn := ⟨(nx.1 + 1), hnx₁⟩
+  have hx₂: fb nx < fb ny := by
+    refine hfb₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fb ny ∈ sb := by
+    rw [hsb₀]
+    exact Set.mem_range_self ny
+  have hx₄: fb ny ≤ br := by
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    rw [hfr]
+    use (fb ny)
+  have hc₁: br < fb ny := by
+    rw [ ← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false br).mp ?_
+  exact lt_of_lt_of_le hc₁ hx₄
+
+lemma imo_1985_p6_bonus_3_2
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (x : NNReal)
+  (hx₀ : x ∈ sb)
+  (hx₁ : fr x = br):
+   False := by
+  rw [hsb₀] at hx₀
+  apply Set.mem_range.mp at hx₀
+  obtain ⟨nx, hnx₀⟩ := hx₀
+  have hnx₁: (nx.1 + (1:ℕ)) ∈ sn := by
+    let t : ℕ := nx.1 + 1
+    have ht₀: t = nx.1 + 1 := by rfl
+    rw [← ht₀, hsn]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_left 1 ↑nx
+  let ny : ↑sn := ⟨(nx.1 + 1), hnx₁⟩
+  have hx₂: fb nx < fb ny := by
+    refine hfb₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fb ny ∈ sb := by
+    rw [hsb₀]
+    exact Set.mem_range_self ny
+  have hx₄: fb ny ≤ br := by
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    rw [hfr]
+    use (fb ny)
+  have hc₁: br < fb ny := by
+    rw [ ← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false br).mp ?_
+  exact lt_of_lt_of_le hc₁ hx₄
+
+
+lemma imo_1985_p6_bonus_3_3
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (x : NNReal)
+  (hx₁ : fr x = br)
+  (nx : ↑sn)
+  (hnx₀ : fb nx = x):
+   False := by
+  have hnx₁: (nx.1 + (1:ℕ)) ∈ sn := by
+    let t : ℕ := nx.1 + 1
+    have ht₀: t = nx.1 + 1 := by rfl
+    rw [← ht₀, hsn]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_left 1 ↑nx
+  let ny : ↑sn := ⟨(nx.1 + 1), hnx₁⟩
+  have hx₂: fb nx < fb ny := by
+    refine hfb₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fb ny ∈ sb := by
+    rw [hsb₀]
+    exact Set.mem_range_self ny
+  have hx₄: fb ny ≤ br := by
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    rw [hfr]
+    use (fb ny)
+  have hc₁: br < fb ny := by
+    rw [ ← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false br).mp ?_
+  exact lt_of_lt_of_le hc₁ hx₄
+
+lemma imo_1985_p6_bonus_3_4
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (nx : ↑sn):
+  nx.1 + 1 ∈ sn := by
+  let t : ℕ := nx.1 + 1
+  have ht₀: t = nx.1 + 1 := by rfl
+  rw [← ht₀, hsn]
+  refine Set.mem_Ici.mpr ?_
+  exact Nat.le_add_left 1 ↑nx
+
+lemma imo_1985_p6_bonus_3_5
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (x : NNReal)
+  (hx₁ : fr x = br)
+  (nx : ↑sn)
+  (hnx₀ : fb nx = x)
+  (hnx₁ : nx.1 + 1 ∈ sn)
+  (ny : ↑sn)
+  (hny : ny = ⟨↑nx + 1, hnx₁⟩) :
+   False := by
+  have hx₂: fb nx < fb ny := by
+    refine hfb₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hny]
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fb ny ∈ sb := by
+    rw [hsb₀]
+    exact Set.mem_range_self ny
+  have hx₄: fb ny ≤ br := by
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    rw [hfr]
+    use (fb ny)
+  have hc₁: br < fb ny := by
+    rw [ ← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false br).mp ?_
+  exact lt_of_lt_of_le hc₁ hx₄
+
+
+lemma imo_1985_p6_bonus_3_6
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (nx : ↑sn)
+  (hnx₁ : nx.1 + 1 ∈ sn)
+  (ny : ↑sn)
+  (hny : ny = ⟨↑nx + 1, hnx₁⟩):
+   fb nx < fb ny := by
+  refine hfb₃ ?_
+  refine Subtype.mk_lt_mk.mpr ?_
+  rw [hny]
+  exact lt_add_one (↑nx:ℕ)
+
+lemma imo_1985_p6_bonus_3_7
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (x : NNReal)
+  (hx₁ : fr x = br)
+  (nx : ↑sn)
+  (hnx₀ : fb nx = x)
+  (ny : ↑sn)
+  (hx₂ : fb nx < fb ny):
+   False := by
+  have hx₃: fb ny ∈ sb := by
+    rw [hsb₀]
+    exact Set.mem_range_self ny
+  have hx₄: fb ny ≤ br := by
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    rw [hfr]
+    use (fb ny)
+  have hc₁: br < fb ny := by
+    rw [ ← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false br).mp ?_
+  exact lt_of_lt_of_le hc₁ hx₄
+
+lemma imo_1985_p6_bonus_3_8
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (x : NNReal)
+  (hx₁ : fr x = br)
+  (nx : ↑sn)
+  (hnx₀ : fb nx = x)
+  (ny : ↑sn)
+  (hx₂ : fb nx < fb ny)
+  (hx₃ : fb ny ∈ sb):
+   False := by
+  have hx₄: fb ny ≤ br := by
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    rw [hfr]
+    use (fb ny)
+  have hc₁: br < fb ny := by
+    rw [ ← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false br).mp ?_
+  exact lt_of_lt_of_le hc₁ hx₄
+
+lemma imo_1985_p6_bonus_3_9
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (ny : ↑sn)
+  (hx₃ : fb ny ∈ sb):
+   ↑(fb ny) ≤ br := by
+  refine hbr₃ _ ?_
+  rw [hsbr]
+  refine (Set.mem_image fr sb _).mpr ?_
+  rw [hfr]
+  use (fb ny)
+
+lemma imo_1985_p6_bonus_3_10
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (ny : ↑sn)
+  (hx₃ : fb ny ∈ sb):
+   ↑(fb ny) ∈ fr '' sb := by
+  refine (Set.mem_image fr sb _).mpr ?_
+  rw [hfr]
+  use (fb ny)
+
+lemma imo_1985_p6_bonus_3_11
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (br : ℝ)
+  (x : NNReal)
+  (hx₁ : fr x = br)
+  (nx : ↑sn)
+  (hnx₀ : fb nx = x)
+  (ny : ↑sn)
+  (hx₂ : fb nx < fb ny)
+  (hx₄ : ↑(fb ny) ≤ br):
+   False := by
+  have hc₁: br < fb ny := by
+    rw [ ← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false br).mp ?_
+  exact lt_of_lt_of_le hc₁ hx₄
+
+
+
+
+
+
+lemma imo_1985_p6_bonus_4
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x) :
+  cr ∉ scr := by
+  rw [hscr]
+  by_contra! hc₀
+  apply (Set.mem_image fr sc cr).mp at hc₀
+  obtain ⟨x, hx₀, hx₁⟩ := hc₀
+  rw [hsc₀] at hx₀
+  apply Set.mem_range.mp at hx₀
+  obtain ⟨nx, hnx₀⟩ := hx₀
+  have hnx₁: nx.1 + 1 ∈ sn := by
+    let t : ℕ := nx.1 + 1
+    have ht₀: t = nx.1 + 1 := by rfl
+    rw [← ht₀, hsn]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_left 1 ↑nx
+  let ny : ↑sn := ⟨(nx.1 + 1), hnx₁⟩
+  have hx₂: fc ny < fc nx := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fc ny ∈ sc := by
+    rw [hsc₀]
+    exact Set.mem_range_self ny
+  have hx₄: cr ≤ fc ny := by
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    rw [hfr]
+    use (fc ny)
+  have hc₁: fc ny < cr := by
+    rw [← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false cr).mp ?_
+  exact lt_of_le_of_lt hx₄ hc₁
+
+
+lemma imo_1985_p6_bonus_4_1
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (hc₀ : cr ∈ fr '' sc):
+   False := by
+  apply (Set.mem_image fr sc cr).mp at hc₀
+  obtain ⟨x, hx₀, hx₁⟩ := hc₀
+  rw [hsc₀] at hx₀
+  apply Set.mem_range.mp at hx₀
+  obtain ⟨nx, hnx₀⟩ := hx₀
+  have hnx₁: nx.1 + 1 ∈ sn := by
+    let t : ℕ := nx.1 + 1
+    have ht₀: t = nx.1 + 1 := by rfl
+    rw [← ht₀, hsn]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_left 1 ↑nx
+  let ny : ↑sn := ⟨(nx.1 + 1), hnx₁⟩
+  have hx₂: fc ny < fc nx := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fc ny ∈ sc := by
+    rw [hsc₀]
+    exact Set.mem_range_self ny
+  have hx₄: cr ≤ fc ny := by
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    rw [hfr]
+    use (fc ny)
+  have hc₁: fc ny < cr := by
+    rw [← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false cr).mp ?_
+  exact lt_of_le_of_lt hx₄ hc₁
+
+lemma imo_1985_p6_bonus_4_2
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (x : NNReal)
+  (hx₀ : x ∈ sc)
+  (hx₁ : fr x = cr):
+   False := by
+  rw [hsc₀] at hx₀
+  apply Set.mem_range.mp at hx₀
+  obtain ⟨nx, hnx₀⟩ := hx₀
+  have hnx₁: nx.1 + 1 ∈ sn := by
+    let t : ℕ := nx.1 + 1
+    have ht₀: t = nx.1 + 1 := by rfl
+    rw [← ht₀, hsn]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_left 1 ↑nx
+  let ny : ↑sn := ⟨(nx.1 + 1), hnx₁⟩
+  have hx₂: fc ny < fc nx := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fc ny ∈ sc := by
+    rw [hsc₀]
+    exact Set.mem_range_self ny
+  have hx₄: cr ≤ fc ny := by
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    rw [hfr]
+    use (fc ny)
+  have hc₁: fc ny < cr := by
+    rw [← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false cr).mp ?_
+  exact lt_of_le_of_lt hx₄ hc₁
+
+lemma imo_1985_p6_bonus_4_3
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (x : NNReal)
+  (hx₁ : fr x = cr)
+  (hx₀ : ∃ y, fc y = x):
+   False := by
+  obtain ⟨nx, hnx₀⟩ := hx₀
+  have hnx₁: nx.1 + 1 ∈ sn := by
+    let t : ℕ := nx.1 + 1
+    have ht₀: t = nx.1 + 1 := by rfl
+    rw [← ht₀, hsn]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_left 1 ↑nx
+  let ny : ↑sn := ⟨(nx.1 + 1), hnx₁⟩
+  have hx₂: fc ny < fc nx := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fc ny ∈ sc := by
+    rw [hsc₀]
+    exact Set.mem_range_self ny
+  have hx₄: cr ≤ fc ny := by
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    rw [hfr]
+    use (fc ny)
+  have hc₁: fc ny < cr := by
+    rw [← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false cr).mp ?_
+  exact lt_of_le_of_lt hx₄ hc₁
+
+lemma imo_1985_p6_bonus_4_4
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (x : NNReal)
+  (hx₁ : fr x = cr)
+  (nx : ↑sn)
+  (hnx₀ : fc nx = x):
+   False := by
+  have hnx₁: nx.1 + 1 ∈ sn := by
+    let t : ℕ := nx.1 + 1
+    have ht₀: t = nx.1 + 1 := by rfl
+    rw [← ht₀, hsn]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_left 1 ↑nx
+  let ny : ↑sn := ⟨(nx.1 + 1), hnx₁⟩
+  have hx₂: fc ny < fc nx := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fc ny ∈ sc := by
+    rw [hsc₀]
+    exact Set.mem_range_self ny
+  have hx₄: cr ≤ fc ny := by
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    rw [hfr]
+    use (fc ny)
+  have hc₁: fc ny < cr := by
+    rw [← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false cr).mp ?_
+  exact lt_of_le_of_lt hx₄ hc₁
+
+
+lemma imo_1985_p6_bonus_4_5
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (nx : ↑sn):
+  nx.1 + 1 ∈ sn := by
+  let t : ℕ := nx.1 + 1
+  have ht₀: t = nx.1 + 1 := by rfl
+  rw [← ht₀, hsn]
+  refine Set.mem_Ici.mpr ?_
+  exact Nat.le_add_left 1 ↑nx
+
+lemma imo_1985_p6_bonus_4_6
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (x : NNReal)
+  (hx₁ : fr x = cr)
+  (nx : ↑sn)
+  (hnx₀ : fc nx = x)
+  (hnx₁ : nx.1 + 1 ∈ sn)
+  (ny : ↑sn)
+  (hny : ny = ⟨↑nx + 1, hnx₁⟩):
+   False := by
+  have hx₂: fc ny < fc nx := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hny]
+    exact lt_add_one (↑nx:ℕ)
+  have hx₃: fc ny ∈ sc := by
+    rw [hsc₀]
+    exact Set.mem_range_self ny
+  have hx₄: cr ≤ fc ny := by
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    rw [hfr]
+    use (fc ny)
+  have hc₁: fc ny < cr := by
+    rw [← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false cr).mp ?_
+  exact lt_of_le_of_lt hx₄ hc₁
+
+lemma imo_1985_p6_bonus_4_7
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (nx : ↑sn)
+  (hnx₁ : nx.1 + 1 ∈ sn)
+  (ny : ↑sn)
+  (hny : ny = ⟨↑nx + 1, hnx₁⟩):
+   fc ny < fc nx := by
+  refine hfc₃ ?_
+  refine Subtype.mk_lt_mk.mpr ?_
+  rw [hny]
+  exact lt_add_one (↑nx:ℕ)
+
+lemma imo_1985_p6_bonus_4_8
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (x : NNReal)
+  (hx₁ : fr x = cr)
+  (nx : ↑sn)
+  (hnx₀ : fc nx = x)
+  (ny : ↑sn)
+  (hx₂ : fc ny < fc nx):
+   False := by
+  have hx₃: fc ny ∈ sc := by
+    rw [hsc₀]
+    exact Set.mem_range_self ny
+  have hx₄: cr ≤ fc ny := by
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    rw [hfr]
+    use (fc ny)
+  have hc₁: fc ny < cr := by
+    rw [← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false cr).mp ?_
+  exact lt_of_le_of_lt hx₄ hc₁
+
+lemma imo_1985_p6_bonus_4_9
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (x : NNReal)
+  (hx₁ : fr x = cr)
+  (nx : ↑sn)
+  (hnx₀ : fc nx = x)
+  (ny : ↑sn)
+  (hx₂ : fc ny < fc nx)
+  (hx₃ : fc ny ∈ sc):
+   False := by
+  have hx₄: cr ≤ fc ny := by
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    rw [hfr]
+    use (fc ny)
+  have hc₁: fc ny < cr := by
+    rw [← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false cr).mp ?_
+  exact lt_of_le_of_lt hx₄ hc₁
+
+lemma imo_1985_p6_bonus_4_10
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (ny : ↑sn)
+  (hx₃ : fc ny ∈ sc):
+   cr ≤ ↑(fc ny) := by
+  refine hcr₃ _ ?_
+  rw [hscr]
+  refine (Set.mem_image fr sc _).mpr ?_
+  rw [hfr]
+  use (fc ny)
+
+lemma imo_1985_p6_bonus_4_11
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (cr : ℝ)
+  (x : NNReal)
+  (hx₁ : fr x = cr)
+  (nx : ↑sn)
+  (hnx₀ : fc nx = x)
+  (ny : ↑sn)
+  (hx₂ : fc ny < fc nx)
+  (hx₄ : cr ≤ ↑(fc ny)):
+   False := by
+  have hc₁: fc ny < cr := by
+    rw [← hx₁, ← hnx₀, hfr]
+    exact hx₂
+  refine (lt_self_iff_false cr).mp ?_
+  exact lt_of_le_of_lt hx₄ hc₁
+
+
+
+lemma imo_1985_p6_bonus_5
+  (f₀ : ℕ → NNReal → NNReal)
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hsn₁ : ∀ (n : ↑sn), ↑n ∈ sn ∧ 0 < n.1)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1) :
+  ∀ (n : ↑sn), f₀ (↑n) (fc n) - f₀ (↑n) (fb n) = 1 / ↑↑n := by
+  intros n
+  have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+  rw [hfb₁ n, hfc₁ n]
+  rw [NNReal.sub_def, NNReal.sub_def]
+  norm_cast
+  simp
+  have g₀: 0 ≤ (1 - (↑n:ℝ)⁻¹) := by
+    refine sub_nonneg.mpr ?_
+    refine inv_le_one_of_one_le₀ ?_
+    exact Nat.one_le_cast.mpr hn₀
+  have g₁: max (1 - (↑n:ℝ)⁻¹) 0 = 1 - (↑n:ℝ)⁻¹ := by
+    exact max_eq_left g₀
+  rw [g₁, ← sub_add, sub_self, zero_add]
+  rw [Real.toNNReal_inv]
+  refine inv_inj.mpr ?_
+  exact NNReal.toNNReal_coe_nat n
+
+
+
+lemma imo_1985_p6_bonus_5_1
+  (f₀ : ℕ → NNReal → NNReal)
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1):
+   f₀ (↑n) (fc n) - f₀ (↑n) (fb n) = 1 / ↑↑n := by
+  rw [hfb₁ n, hfc₁ n]
+  rw [NNReal.sub_def, NNReal.sub_def]
+  norm_cast
+  simp
+  have g₀: 0 ≤ (1 - (↑n:ℝ)⁻¹) := by
+    refine sub_nonneg.mpr ?_
+    refine inv_le_one_of_one_le₀ ?_
+    exact Nat.one_le_cast.mpr hn₀
+  have g₁: max (1 - (↑n:ℝ)⁻¹) 0 = 1 - (↑n:ℝ)⁻¹ := by
+    exact max_eq_left g₀
+  rw [g₁, ← sub_add, sub_self, zero_add]
+  rw [Real.toNNReal_inv]
+  refine inv_inj.mpr ?_
+  exact NNReal.toNNReal_coe_nat n
+
+
+lemma imo_1985_p6_bonus_5_2
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1):
+  ((1:NNReal).toReal - ↑((1:NNReal).toReal - ↑((1) / ↑↑n)).toNNReal).toNNReal = (1:NNReal) / ↑↑n := by
+  norm_cast
+  simp
+  have g₀: 0 ≤ (1 - (↑n:ℝ)⁻¹) := by
+    refine sub_nonneg.mpr ?_
+    refine inv_le_one_of_one_le₀ ?_
+    exact Nat.one_le_cast.mpr hn₀
+  have g₁: max (1 - (↑n:ℝ)⁻¹) 0 = 1 - (↑n:ℝ)⁻¹ := by
+    exact max_eq_left g₀
+  rw [g₁, ← sub_add, sub_self, zero_add]
+  rw [Real.toNNReal_inv]
+  refine inv_inj.mpr ?_
+  exact NNReal.toNNReal_coe_nat n
+
+lemma imo_1985_p6_bonus_5_3
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1):
+   ((1:ℝ) - (1 - (↑↑n)⁻¹) ⊔ 0).toNNReal = (↑↑n)⁻¹ := by
+  have g₀: 0 ≤ (1 - (↑n:ℝ)⁻¹) := by
+    refine sub_nonneg.mpr ?_
+    refine inv_le_one_of_one_le₀ ?_
+    exact Nat.one_le_cast.mpr hn₀
+  have g₁: max (1 - (↑n:ℝ)⁻¹) 0 = 1 - (↑n:ℝ)⁻¹ := by
+    exact max_eq_left g₀
+  rw [g₁, ← sub_add, sub_self, zero_add]
+  rw [Real.toNNReal_inv]
+  refine inv_inj.mpr ?_
+  exact NNReal.toNNReal_coe_nat n
+
+lemma imo_1985_p6_bonus_5_4
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1):
+   (0:ℝ) ≤ 1 - (↑↑n)⁻¹ := by
+  refine sub_nonneg.mpr ?_
+  refine inv_le_one_of_one_le₀ ?_
+  exact Nat.one_le_cast.mpr hn₀
+
+lemma imo_1985_p6_bonus_5_5
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (g₀ : (0:ℝ) ≤ 1 - (↑↑n)⁻¹):
+   ((1:ℝ) - (1 - (↑↑n)⁻¹) ⊔ 0).toNNReal = (↑↑n)⁻¹ := by
+  have g₁: max (1 - (↑n:ℝ)⁻¹) 0 = 1 - (↑n:ℝ)⁻¹ := by
+    exact max_eq_left g₀
+  rw [g₁, ← sub_add, sub_self, zero_add]
+  rw [Real.toNNReal_inv]
+  refine inv_inj.mpr ?_
+  exact NNReal.toNNReal_coe_nat n
+
+
+lemma imo_1985_p6_2
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x) :
+  ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y := by
+  intros n x y hn hxy
+  by_cases hn₁: 1 < n
+  . refine Nat.le_induction ?_ ?_ n hn₁
+    . rw [h₁ 1 x (by norm_num)]
+      rw [h₁ 1 y (by norm_num)]
+      norm_num
+      refine mul_lt_mul ?_ ?_ ?_ ?_
+      . rw [h₀ x, h₀ y]
+        exact hxy
+      . refine _root_.add_le_add ?_ (by norm_num)
+        rw [h₀ x, h₀ y]
+        exact le_of_lt hxy
+      . refine add_pos_of_nonneg_of_pos ?_ (by linarith)
+        rw [h₀ x]
+        exact NNReal.zero_le_coe
+      . refine le_of_lt ?_
+        refine h₂ 1 y ?_
+        norm_num
+        exact pos_of_gt hxy
+    . intros m hm₀ hm₁
+      rw [h₁ m x (by linarith)]
+      rw [h₁ m y (by linarith)]
+      refine mul_lt_mul hm₁ ?_ ?_ ?_
+      . refine _root_.add_le_add ?_ (by norm_num)
+        refine le_of_lt hm₁
+      . refine add_pos_of_nonneg_of_pos ?_ ?_
+        . exact h₃ m x (by linarith)
+        . refine one_div_pos.mpr ?_
+          norm_cast
+          exact Nat.zero_lt_of_lt hm₀
+      . refine le_of_lt ?_
+        refine h₂ m y ?_
+        constructor
+        . exact Nat.zero_lt_of_lt hm₀
+        . exact pos_of_gt hxy
+  . interval_cases n
+    rw [h₀ x, h₀ y]
+    exact hxy
+
+
+lemma imo_1985_p6_4
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x => (f n x).toNNReal) :
+  ∀ (n : ℕ), 0 < n → StrictMono (f₀ n) := by
+  intros n hn₀
+  refine Monotone.strictMono_of_injective ?_ ?_
+  . refine monotone_iff_forall_lt.mpr ?_
+    intros a b hab
+    refine le_of_lt ?_
+    rw [hf₀]
+    exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n a hn₀)).mpr (h₄ n a b hn₀ hab)
+  . intros p q hpq
+    contrapose! hpq
+    apply lt_or_gt_of_ne at hpq
+    cases' hpq with hpq hpq
+    . refine ne_of_lt ?_
+      rw [hf₀]
+      exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n p hn₀)).mpr (h₄ n p q hn₀ hpq)
+    . symm
+      refine ne_of_lt ?_
+      rw [hf₀]
+      exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n q hn₀)).mpr (h₄ n q p hn₀ hpq)
+
+lemma imo_1985_p6_5
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n => Function.invFun (f₀ n)):
+  ∀ (n : ℕ) (x y : NNReal), 0 < n → f₀ n x = y → fi n y = x := by
+  intros n x y hn₀ hn₁
+  have hf₃: ∀ n y, fi n y = Function.invFun (f₀ n) y := by
+    exact fun n y => congrFun (congrFun hfi n) y
+  rw [← hn₁, hf₃]
+  have hmo₃: ∀ n, 0 < n → Function.Injective (f₀ n) := by
+    exact fun n a => StrictMono.injective (hmo₂ n a)
+  have hn₂: (Function.invFun (f₀ n)) ∘ (f₀ n) = id := by exact Function.invFun_comp (hmo₃ n hn₀)
+  rw [Function.comp_def (Function.invFun (f₀ n)) (f₀ n)] at hn₂
+  have hn₃: (fun x => Function.invFun (f₀ n) (f₀ n x)) x = id x := by exact Eq.symm (NNReal.eq (congrArg NNReal.toReal (congrFun (id (Eq.symm hn₂)) x)))
+  exact hmo₁ n hn₀ (congrArg (f n) hn₃)
+
+lemma imo_1985_p6_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x => (f n x).toNNReal) :
+  ∀ (n : ℕ), 0 < n → Continuous (f₀ n) := by
+  intros n hn₀
+  rw [hf₀]
+  refine Continuous.comp' ?_ ?_
+  . exact continuous_real_toNNReal
+  . refine Nat.le_induction ?_ ?_ n hn₀
+    . have hn₁: f 1 = fun (x:NNReal) => (x:ℝ) := by exact (Set.eqOn_univ (f 1) fun x => ↑x).mp fun ⦃x⦄ _ => h₀ x
+      rw [hn₁]
+      exact NNReal.continuous_coe
+    . intros d hd₀ hd₁
+      have hd₂: f (d + 1) = fun x => f d x * (f d x + 1 / ↑d) := by
+        exact (Set.eqOn_univ (f (d + 1)) fun x => f d x * (f d x + 1 / ↑d)).mp fun ⦃x⦄ _ => h₁ d x hd₀
+      rw [hd₂]
+      refine Continuous.mul hd₁ ?_
+      refine Continuous.add hd₁ ?_
+      exact continuous_const
+
+lemma imo_1985_p6_11
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (hcr₀ : IsGLB scr cr)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n) :
+  br ≤ cr := by
+  have hfc₄: ∀ nb nc, fb nb < fc nc := by
+    intros nb nc
+    cases' (lt_or_le nb nc) with hn₀ hn₀
+    . refine lt_trans ?_ (hfc₂ nc)
+      exact hfb₃ hn₀
+    cases' lt_or_eq_of_le hn₀ with hn₁ hn₁
+    . refine lt_trans (hfc₂ nb) ?_
+      exact hfc₃ hn₁
+    . rw [hn₁]
+      exact hfc₂ nb
+  by_contra! hc₀
+  have hc₁: ∃ x ∈ sbr, cr < x ∧ x ≤ br := by exact IsLUB.exists_between hbr₀ hc₀
+  let ⟨x, hx₀, hx₁, _⟩ := hc₁
+  have hc₂: ∃ y ∈ scr, cr ≤ y ∧ y < x := by exact IsGLB.exists_between hcr₀ hx₁
+  let ⟨y, hy₀, _, hy₂⟩ := hc₂
+  have hc₃: x < y := by
+    have hx₃: x.toNNReal ∈ sb := by
+      rw [hsbr] at hx₀
+      apply (Set.mem_image fr sb x).mp at hx₀
+      obtain ⟨z, hz₀, hz₁⟩ := hx₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    have hy₃: y.toNNReal ∈ sc := by
+      rw [hscr] at hy₀
+      apply (Set.mem_image fr sc y).mp at hy₀
+      obtain ⟨z, hz₀, hz₁⟩ := hy₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    rw [hsb₀] at hx₃
+    rw [hsc₀] at hy₃
+    apply Set.mem_range.mp at hx₃
+    apply Set.mem_range.mp at hy₃
+    let ⟨nx, hnx₀⟩ := hx₃
+    let ⟨ny, hny₀⟩ := hy₃
+    have hy₄: 0 < y := by
+      contrapose! hy₃
+      have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+      intro z
+      rw [hy₅]
+      refine ne_of_gt ?_
+      refine lt_of_le_of_lt ?_ (hfc₂ z)
+      exact hfb₄ z
+    refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+    rw [← hnx₀, ← hny₀]
+    exact hfc₄ nx ny
+  refine (lt_self_iff_false x).mp ?_
+  exact lt_trans hc₃ hy₂
+
+lemma imo_1985_p6_9
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hf₅ : ∀ (x : NNReal), fi 1 x = x)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (fb : ℕ → NNReal)
+  (hfb₀ : fb = fun n => fi n (1 - 1 / ↑n))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1) :
+  StrictMonoOn fb sn := by
+  rw [hsn]
+  refine strictMonoOn_Ici_of_pred_lt ?hψ
+  intros m hm₀
+  rw [hfb₀]
+  refine Nat.le_induction ?_ ?_ m hm₀
+  . have g₁: fi 1 0 = 0 := by exact hf₅ 0
+    have g₂: (2:NNReal).IsConjExponent (2:NNReal) := by
+      refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+      . exact one_lt_two
+      . norm_cast
+        simp
+    simp
+    norm_cast
+    rw [g₁, NNReal.IsConjExponent.one_sub_inv g₂]
+    let x := fi 2 2⁻¹
+    have hx₀: x = fi 2 2⁻¹ := by rfl
+    have hx₁: f₀ 2 x = 2⁻¹ := by
+      rw [hx₀]
+      have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+      exact g₃ 2⁻¹
+    rw [← hx₀]
+    contrapose! hx₁
+    have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+    have hc₃: f₀ 2 x = 0 := by
+      rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+      norm_cast
+      rw [zero_mul]
+      exact Real.toNNReal_zero
+    rw [hc₃]
+    exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+  . simp
+    intros n hn₀ _
+    let i := fi n (1 - (↑n)⁻¹)
+    let j := fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹)
+    have hi₀: i = fi n (1 - (↑n)⁻¹) := by rfl
+    have hj₀: j = fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹) := by rfl
+    have hi₁: f₀ n i = (1 - (↑n)⁻¹) := by exact (hf₇ n i (1 - (↑n:NNReal)⁻¹) (by linarith)).mpr hi₀.symm
+    have hj₁: f₀ (n + 1) j = (1 - ((↑n:NNReal) + 1)⁻¹) := by
+      exact (hf₇ (n + 1) j _ (by linarith)).mpr hj₀.symm
+    have hj₂: (1 - ((↑n:NNReal) + 1)⁻¹) = (1 - ((n:ℝ) + 1)⁻¹).toNNReal := by
+      exact rfl
+    have hn₂: f₀ (n + 1) i < f₀ (n + 1) j := by
+      rw [hj₁, hj₂, hf₂ (n + 1) _ (by linarith), h₁ n i (by linarith)]
+      rw [hf₁ n i (by linarith), hi₁]
+      refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+      . refine sub_pos.mpr ?_
+        refine inv_lt_one_of_one_lt₀ ?_
+        norm_cast
+        exact Nat.lt_add_right 1 hn₀
+      . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+        rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+        simp
+        refine inv_strictAnti₀ ?_ ?_
+        . norm_cast
+          exact Nat.zero_lt_of_lt hn₀
+        . norm_cast
+          exact lt_add_one n
+    refine (StrictMono.lt_iff_lt ?_).mp hn₂
+    exact hmo₂ (n + 1) (by linarith)
+
+
+lemma imo_1985_p6_10
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (sb : Set NNReal)
+  (fb : ↑sn → NNReal)
+  (hsn₀ : sn = Set.Ici 1)
+  (hfb₀ : fb = fun n:↑sn => fi (↑n) (1 - 1 / ↑↑n))
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr: fr = fun x => ↑x)
+  (sbr : Set ℝ)
+  (hsbr: sbr = fr '' sb)
+  (br: ℝ)
+  (hbr₀ : IsLUB sbr br) :
+  0 < br := by
+  have hnb₀: 2 ∈ sn := by
+    rw [hsn₀]
+    decide
+  let nb : ↑sn := ⟨2, hnb₀⟩
+  have g₀: 0 < fb nb := by
+    have g₁: (2:NNReal).IsConjExponent (2:NNReal) := by
+      refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+      . exact one_lt_two
+      . norm_cast
+        simp
+    rw [hfb₀]
+    simp
+    have hnb₁: nb.val = 2 := by exact rfl
+    rw [hnb₁]
+    norm_cast
+    rw [NNReal.IsConjExponent.one_sub_inv g₁]
+    let x := fi 2 2⁻¹
+    have hx₀: x = fi 2 2⁻¹ := by rfl
+    have hx₁: f₀ 2 x = 2⁻¹ := by
+      rw [hx₀]
+      have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+      exact g₃ 2⁻¹
+    rw [← hx₀]
+    contrapose! hx₁
+    have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+    have hc₃: f₀ 2 x = 0 := by
+      rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+      norm_cast
+      rw [zero_mul]
+      exact Real.toNNReal_zero
+    rw [hc₃]
+    exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₁)
+  have g₁: ∃ x, 0 < x ∧ x ∈ sbr := by
+    use (fb nb).toReal
+    constructor
+    . exact g₀
+    . rw [hsbr]
+      simp
+      use fb ↑nb
+      constructor
+      . rw [hsb₀]
+        exact Set.mem_range_self nb
+      . exact congrFun hfr (fb ↑nb)
+  obtain ⟨x, hx₀, hx₁⟩ := g₁
+  have hx₂: br ∈ upperBounds sbr := by
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  exact gt_of_ge_of_gt (hx₂ hx₁) hx₀
+
+
+lemma imo_1985_p6_unique
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x) :
+  ∀ (y₁ y₂ : NNReal),
+    (∀ (n : ℕ), 0 < n → 0 < f n y₁ ∧ f n y₁ < f (n + 1) y₁ ∧ f (n + 1) y₁ < 1) →
+      (∀ (n : ℕ), 0 < n → 0 < f n y₂ ∧ f n y₂ < f (n + 1) y₂ ∧ f (n + 1) y₂ < 1) → y₁ = y₂ := by
+  intros x y hx₀ hy₀
+  let sd : Set ℕ := Set.Ici 2
+  let fd : NNReal → NNReal → ↑sd → ℝ := fun y₁ y₂ n => (f n.1 y₂ - f n.1 y₁)
+  have hfd₁: ∀ y₁ y₂ n, fd y₁ y₂ n = f n.1 y₂ - f n.1 y₁ := by exact fun y₁ y₂ n => rfl
+  have hd₁: ∀ n a b, a < b → 0 < fd a b n := by
+    intros nd a b hnd₀
+    rw [hfd₁]
+    refine sub_pos.mpr ?_
+    refine hmo₀ nd.1 ?_ hnd₀
+    exact lt_of_lt_of_le (Nat.zero_lt_two) nd.2
+  have hfd₂: ∀ a b, a < b → (∀ n:↑sd, f n.1 a < f (n.1 + 1) a ∧ f n.1 b < f (n.1 + 1) b)
+      → Filter.Tendsto (fd a b) Filter.atTop Filter.atTop := by
+    intros a b ha₀ ha₁
+    have hd₀: ∀ nd:↑sd, (nd.1 + 1) ∈ sd := by
+      intro nd
+      have hd₀: 2 ≤ nd.1 := by exact nd.2
+      refine Set.mem_Ici.mpr ?_
+      exact Nat.le_add_right_of_le hd₀
+    have hd₂: ∀ nd, fd a b nd  * (2 - 1 / nd.1) ≤ fd a b ⟨nd.1 + 1, hd₀ nd⟩ := by
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact Nat.lt_add_left_iff_pos.mp (hd₀ nd)
+      rw [hfd₁, hfd₁, h₁ nd.1 _ hnd₀, h₁ nd.1 _ hnd₀]
+      have hnd₁: f (↑nd) b * (f (↑nd) b + 1 / ↑↑nd) - f (↑nd) a * (f (↑nd) a + 1 / ↑↑nd) =
+        (f (↑nd) b - f (↑nd) a) * (f (↑nd) b + f (↑nd) a + 1 / nd.1) := by
+        ring_nf
+      rw [hnd₁]
+      refine (mul_le_mul_left ?_).mpr ?_
+      . rw [← hfd₁]
+        exact hd₁ nd a b ha₀
+      . refine le_sub_iff_add_le.mp ?_
+        rw [sub_neg_eq_add]
+        have hnd₂: 1 - 1 / nd.1 < f (↑nd) b := by
+          exact h₇ nd.1 b hnd₀ (ha₁ nd).2
+        have hnd₃: 1 - 1 / nd.1 < f (↑nd) a := by
+          exact h₇ nd.1 a hnd₀ (ha₁ nd).1
+        linarith
+    let i : ↑sd := ⟨2, (by decide)⟩
+    have hd₃: ∀ nd, fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd := by
+      intro nd
+      induction' nd with nd hnd₀
+      refine Nat.le_induction ?_ ?_ nd hnd₀
+      . simp
+        exact le_of_eq (by rfl)
+      . simp
+        intros n hn₀ hn₁
+        have hn₂: n - 1 = n - 2 + 1 := by
+          simp
+          exact (Nat.sub_eq_iff_eq_add hn₀).mp rfl
+        refine le_trans ?_ (hd₂ ⟨n, hn₀⟩)
+        rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+        refine mul_le_mul hn₁ ?_ (by linarith) ?_
+        . refine (div_le_iff₀ (two_pos)).mpr ?_
+          rw [sub_mul, one_div_mul_eq_div _ 2]
+          refine le_sub_iff_add_le.mpr ?_
+          refine le_sub_iff_add_le'.mp ?_
+          refine (div_le_iff₀ ?_).mpr ?_
+          . refine Nat.cast_pos.mpr ?_
+            exact lt_of_lt_of_le (two_pos) hn₀
+          . ring_nf
+            exact Nat.ofNat_le_cast.mpr hn₀
+        . exact le_of_lt (hd₁ ⟨n, hn₀⟩ a b ha₀)
+    refine Filter.tendsto_atTop_atTop.mpr ?_
+    intro z
+    by_cases hz₀: z ≤ fd a b i
+    . use i
+      intros j _
+      refine le_trans hz₀ ?_
+      refine le_trans ?_ (hd₃ j)
+      refine le_mul_of_one_le_right ?_ ?_
+      . refine le_of_lt ?_
+        exact hd₁ i a b ha₀
+      . refine one_le_pow₀ ?_
+        linarith
+    . push_neg at hz₀
+      have hz₁: 0 < fd a b i := by exact hd₁ i a b ha₀
+      have hz₂: 0 < Real.log (z / fd a b i) := by
+        refine Real.log_pos ?_
+        exact (one_lt_div hz₁).mpr hz₀
+      let j : ℕ := Nat.ceil (2 + Real.log (z / fd a b i) / Real.log (3 / 2))
+      have hj₀: 2 < j := by
+        refine Nat.lt_ceil.mpr ?_
+        norm_cast
+        refine lt_add_of_pos_right 2 ?_
+        refine div_pos ?_ ?_
+        . exact hz₂
+        . refine Real.log_pos ?_
+          linarith
+      have hj₁: j ∈ sd := by
+        exact Set.mem_Ici_of_Ioi hj₀
+      use ⟨j, hj₁⟩
+      intro k hk₀
+      have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+        exact hd₃ k
+      have hk₂: i < k := by
+        refine lt_of_lt_of_le ?_ hk₀
+        refine Subtype.mk_lt_mk.mpr ?_
+        refine Nat.lt_ceil.mpr ?_
+        norm_cast
+        refine lt_add_of_pos_right 2 ?_
+        refine div_pos ?_ ?_
+        . exact hz₂
+        . refine Real.log_pos ?_
+          linarith
+      refine le_trans ?_ hk₁
+      refine (div_le_iff₀' ?_).mp ?_
+      . exact hz₁
+      . refine Real.le_pow_of_log_le (by linarith) ?_
+        refine (div_le_iff₀ ?_).mp ?_
+        . refine Real.log_pos ?_
+          linarith
+        . rw [Nat.cast_sub ?_]
+          . rw [Nat.cast_two]
+            refine le_sub_iff_add_le'.mpr ?_
+            exact Nat.le_of_ceil_le hk₀
+          . exact Nat.le_trans (hd₀ i) hk₂
+  have hfd₃: ∀ a b, a < b → (∀ (n:↑sd), (1 - 1 / n.1 < f n.1 a ∧ 1 - 1 / n.1 < f n.1 b) ∧ (f n.1 a < 1 ∧ f n.1 b < 1))
+      → Filter.Tendsto (fd a b) Filter.atTop (nhds 0) := by
+    intros a b ha₀ ha₁
+    refine tendsto_atTop_nhds.mpr ?_
+    intros U hU₀ hU₁
+    have hU₂: U ∈ nhds 0 := by exact IsOpen.mem_nhds hU₁ hU₀
+    apply mem_nhds_iff_exists_Ioo_subset.mp at hU₂
+    obtain ⟨l, u, hl₀, hl₁⟩ := hU₂
+    have hl₂: 0 < u := by exact (Set.mem_Ioo.mpr hl₀).2
+    let nd := 2 + Nat.ceil (1/u)
+    have hnd₀: nd ∈ sd := by
+      refine Set.mem_Ici.mpr ?_
+      exact Nat.le_add_right 2 ⌈1 / u⌉₊
+    use ⟨nd, hnd₀⟩
+    intros n hn₀
+    refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+    refine mem_nhds_iff.mpr ?_
+    use Set.Ioo l u
+    constructor
+    . exact hl₁
+    constructor
+    . exact isOpen_Ioo
+    . refine Set.mem_Ioo.mpr ?_
+      constructor
+      . refine lt_trans ?_ (hd₁ n a b ha₀)
+        exact (Set.mem_Ioo.mp hl₀).1
+      . have hn₁: fd a b n < 1 / n := by
+          rw [hfd₁]
+          have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+          have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+          refine sub_lt_iff_lt_add.mpr ?_
+          refine lt_trans hb₁ ?_
+          exact sub_lt_iff_lt_add'.mp ha₂
+        have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+          refine one_div_le_one_div_of_le ?_ ?_
+          . refine Nat.cast_pos.mpr ?_
+            exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+          . exact Nat.cast_le.mpr hn₀
+        refine lt_of_lt_of_le hn₁ ?_
+        refine le_trans hn₂ ?_
+        refine div_le_of_le_mul₀ ?_ ?_ ?_
+        . exact Nat.cast_nonneg' nd
+        . exact le_of_lt hl₂
+        . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+            refine (mul_le_mul_left hl₂).mpr ?_
+            rw [Nat.cast_add 2 _, Nat.cast_two]
+            refine add_le_add_left ?_ 2
+            exact Nat.le_ceil (1 / u)
+          refine le_trans ?_ hl₃
+          rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+          refine le_of_lt ?_
+          refine sub_lt_iff_lt_add.mp ?_
+          rw [sub_self 1]
+          exact mul_pos hl₂ (two_pos)
+  by_contra! hc₀
+  by_cases hy₁: x < y
+  . have hy₂: Filter.Tendsto (fd x y) Filter.atTop Filter.atTop := by
+      refine hfd₂ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) nd.2
+      constructor
+      . exact (hx₀ nd.1 hnd₀).2.1
+      . exact (hy₀ nd.1 hnd₀).2.1
+    have hy₃: Filter.Tendsto (fd x y) Filter.atTop (nhds 0) := by
+      refine hfd₃ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by
+        refine lt_of_lt_of_le ?_ nd.2
+        exact Nat.zero_lt_two
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        refine lt_of_lt_of_le ?_ nd.2
+        exact Nat.one_lt_two
+      constructor
+      . constructor
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₂
+    apply tendsto_atTop_nhds.mp at hy₃
+    contrapose! hy₃
+    clear hy₃
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact N.2
+        . refine le_trans ?_ i.2
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd x y a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+  . have hy₂: y < x := by
+      push_neg at hy₁
+      exact lt_of_le_of_ne hy₁ hc₀.symm
+    have hy₃: Filter.Tendsto (fd y x) Filter.atTop Filter.atTop := by
+      refine hfd₂ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) nd.2
+      constructor
+      . exact (hy₀ nd.1 hnd₀).2.1
+      . exact (hx₀ nd.1 hnd₀).2.1
+    have hy₄: Filter.Tendsto (fd y x) Filter.atTop (nhds 0) := by
+      refine hfd₃ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (Nat.zero_lt_two) nd.2
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        exact lt_of_lt_of_le (Nat.one_lt_two) nd.2
+      constructor
+      . constructor
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₃
+    apply tendsto_atTop_nhds.mp at hy₄
+    contrapose! hy₄
+    clear hy₄
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd y x a := by exact hy₃ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact N.2
+        . refine le_trans ?_ i.2
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd y x a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+
+
+lemma imo_1985_p6_unique_22
+  (f : ℕ → NNReal → ℝ)
+  (x y : NNReal)
+  (sd : Set ℕ := Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ := fun y₁ y₂ n ↦ f (↑n) y₂ - f (↑n) y₁)
+  (sx : Set ℝ)
+  (hsx : sx = Set.Ioo (-1) 1)
+  (N i : ↑sd)
+  (hi₀ : ∀ (a : ↑sd), i ≤ a → ↑↑N + 3 ≤ fd x y a)
+  (hi₁ : N.1 + i.1 ∈ sd)
+  (a : ↑sd)
+  (ha : a = ⟨↑N + ↑i, hi₁⟩):
+   N ≤ a ∧ fd x y a ∉ sx := by
+  constructor
+  . refine Subtype.mk_le_mk.mpr ?_
+    rw [ha]
+    exact Nat.le_add_right ↑N ↑i
+  . rw [hsx]
+    refine Set.not_mem_Ioo_of_ge ?_
+    have hi₂: ↑↑N + 3 ≤ fd x y a := by
+      refine hi₀ a ?_
+      refine Subtype.mk_le_mk.mpr ?_
+      rw [ha]
+      exact Nat.le_add_left ↑i ↑N
+    refine le_trans ?_ hi₂
+    norm_cast
+    exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_exists
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hu₅ : br ≤ cr)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x) :
+  ∃ x, ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1 := by
+  cases' lt_or_eq_of_le hu₅ with hu₆ hu₆
+  . apply exists_between at hu₆
+    let ⟨a, ha₀, ha₁⟩ := hu₆
+    have ha₂: 0 < a := by exact gt_trans ha₀ hbr₁
+    have ha₃: 0 < a.toNNReal := by exact Real.toNNReal_pos.mpr ha₂
+    use a.toNNReal
+    intros n hn₀
+    have hn₁: n ∈ sn := by
+      rw [hsn₀]
+      exact hn₀
+    constructor
+    . exact h₂ n a.toNNReal ⟨hn₀, ha₃⟩
+    constructor
+    . refine h₈ n a.toNNReal hn₀ ?_ ?_
+      . exact Real.toNNReal_pos.mpr ha₂
+      . let nn : ↑sn := ⟨n, hn₁⟩
+        have hn₂: f n (fb nn) = 1 - 1 / n := by
+          rw [hf₁ n _ hn₀, hfb₁ nn]
+          refine NNReal.coe_sub ?_
+          refine div_le_self ?_ ?_
+          . exact zero_le_one' NNReal
+          . exact Nat.one_le_cast.mpr hn₀
+        rw [← hn₂]
+        refine hmo₀ n hn₀ ?_
+        refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+        refine lt_of_le_of_lt ?_ ha₀
+        refine hbr₃ _ ?_
+        rw [hsbr]
+        refine (Set.mem_image fr sb _).mpr ?_
+        use (fb nn)
+        rw [hfr, hsb₀]
+        refine ⟨?_, rfl⟩
+        exact Set.mem_range_self nn
+    . have hn₂: n + 1 ∈ sn := by
+        rw [hsn₀]
+        exact Set.mem_Ici.mpr (by linarith)
+      let nn : ↑sn := ⟨n + 1, hn₂⟩
+      have hn₃: f (n + 1) (fc (nn)) = 1 := by
+        rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+        exact rfl
+      rw [← hn₃]
+      refine hmo₀ (n + 1) (by linarith) ?_
+      refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+      refine lt_of_lt_of_le ha₁ ?_
+      refine hcr₃ _ ?_
+      rw [hscr]
+      refine (Set.mem_image fr sc _).mpr ?_
+      use (fc nn)
+      rw [hfr, hsc₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self nn
+  . use br.toNNReal
+    intros n hn₀
+    have hn₁: n ∈ sn := by
+      rw [hsn₀]
+      exact hn₀
+    constructor
+    . refine h₂ n br.toNNReal ⟨hn₀, ?_⟩
+      exact Real.toNNReal_pos.mpr hbr₁
+    constructor
+    . refine h₈ n br.toNNReal hn₀ ?_ ?_
+      . exact Real.toNNReal_pos.mpr hbr₁
+      . let nn : ↑sn := ⟨n, hn₁⟩
+        have hn₂: fb nn < br := by
+          by_contra! hc₀
+          have hbr₅: (fb nn) = br := by
+            refine eq_of_le_of_le ?_ hc₀
+            refine hbr₃ _ ?_
+            rw [hsbr]
+            refine (Set.mem_image fr sb _).mpr ?_
+            use (fb nn)
+            rw [hfr, hsb₀]
+            constructor
+            . exact Set.mem_range_self nn
+            . exact rfl
+          have hn₂: n + 1 ∈ sn := by
+            rw [hsn₀]
+            refine Set.mem_Ici.mpr ?_
+            exact Nat.le_add_right_of_le hn₀
+          let ns : ↑sn := ⟨n + 1, hn₂⟩
+          have hc₁: fb nn < fb ns := by
+            refine hfb₃ ?_
+            refine Subtype.mk_lt_mk.mpr ?_
+            exact lt_add_one n
+          have hbr₆: fb ns ≤ fb nn := by
+            refine NNReal.coe_le_coe.mp ?_
+            rw [hbr₅]
+            refine hbr₃ _ ?_
+            rw [hsbr]
+            refine (Set.mem_image fr sb _).mpr ?_
+            use (fb ns)
+            rw [hfr, hsb₀]
+            refine ⟨?_, rfl⟩
+            exact Set.mem_range_self ns
+          refine (lt_self_iff_false (fb nn)).mp ?_
+          exact lt_of_lt_of_le hc₁ hbr₆
+        have hn₃: f n (fb nn) = 1 - 1 / n := by
+          rw [hf₁ n _ hn₀, hfb₁ nn]
+          refine NNReal.coe_sub ?_
+          refine div_le_self ?_ ?_
+          . exact zero_le_one' NNReal
+          . exact Nat.one_le_cast.mpr hn₀
+        rw [← hn₃]
+        refine hmo₀ n hn₀ ?_
+        exact Real.lt_toNNReal_iff_coe_lt.mpr hn₂
+    . have hn₂: n + 1 ∈ sn := by
+        rw [hsn₀]
+        exact Set.mem_Ici.mpr (by linarith)
+      let nn : ↑sn := ⟨n + 1, hn₂⟩
+      have hcr₁: 0 < cr := by exact gt_of_ge_of_gt hu₅ hbr₁
+      have hn₃: f (n + 1) (fc (nn)) = 1 := by
+        rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+        exact rfl
+      rw [← hn₃, hu₆]
+      refine hmo₀ (n + 1) (by linarith) ?_
+      refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt hcr₁)).mpr ?_
+      by_contra! hc₀
+      have hc₁: fc nn = cr := by
+        refine eq_of_le_of_le hc₀ ?_
+        refine hcr₃ _ ?_
+        rw [hscr]
+        refine (Set.mem_image fr sc _).mpr ?_
+        use (fc nn)
+        rw [hfr, hsc₀]
+        refine ⟨?_, rfl⟩
+        exact Set.mem_range_self nn
+      have hn₄: n + 2 ∈ sn := by
+        rw [hsn₀]
+        refine Set.mem_Ici.mpr ?_
+        exact Nat.le_add_right_of_le hn₀
+      let ns : ↑sn := ⟨n + 2, hn₄⟩
+      have hn₅: fc ns < fc nn := by
+        refine hfc₃ ?_
+        refine Subtype.mk_lt_mk.mpr ?_
+        exact Nat.lt_add_one (n + 1)
+      have hc₂: fc nn ≤ fc ns := by
+        refine NNReal.coe_le_coe.mp ?_
+        rw [hc₁]
+        refine hcr₃ _ ?_
+        rw [hscr]
+        refine (Set.mem_image fr sc _).mpr ?_
+        use (fc ns)
+        rw [hfr, hsc₀]
+        refine ⟨?_, rfl⟩
+        exact Set.mem_range_self ns
+      refine (lt_self_iff_false (fc ns)).mp ?_
+      exact lt_of_lt_of_le hn₅ hc₂
+
+
+lemma imo_1985_p6_unique_23
+  (f : ℕ → NNReal → ℝ)
+  (x y : NNReal)
+  (sd : Set ℕ := Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ := fun y₁ y₂ n ↦ f (↑n) y₂ - f (↑n) y₁)
+  (sx : Set ℝ)
+  (hsx : sx = Set.Ioo (-1) 1)
+  (N i : ↑sd)
+  (hi₀ : ∀ (a : ↑sd), i ≤ a → ↑↑N + 3 ≤ fd x y a)
+  (hi₁ : N.1 + ↑i ∈ sd)
+  (a : ↑sd)
+  (ha : a = ⟨↑N + ↑i, hi₁⟩):
+   fd x y a ∉ sx := by
+  rw [hsx]
+  refine Set.not_mem_Ioo_of_ge ?_
+  have hi₂: ↑↑N + 3 ≤ fd x y a := by
+    refine hi₀ a ?_
+    refine Subtype.mk_le_mk.mpr ?_
+    rw [ha]
+    exact Nat.le_add_left ↑i ↑N
+  refine le_trans ?_ hi₂
+  norm_cast
+  exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_unique_24
+  (f : ℕ → NNReal → ℝ)
+  (x y : NNReal)
+  (sd : Set ℕ := Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ := fun y₁ y₂ n ↦ f (↑n) y₂ - f (↑n) y₁)
+  (N i : ↑sd)
+  (hi₀ : ∀ (a : ↑sd), i ≤ a → ↑↑N + 3 ≤ fd x y a)
+  (hi₁ : N.1 + ↑i ∈ sd)
+  (a : ↑sd)
+  (ha : a = ⟨↑N + ↑i, hi₁⟩):
+   ↑↑N + 3 ≤ fd x y a := by
+  refine hi₀ a ?_
+  refine Subtype.mk_le_mk.mpr ?_
+  rw [ha]
+  exact Nat.le_add_left ↑i ↑N
+
+
+lemma imo_1985_p6_main_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have h₃: ∀ n x, 0 < n → 0 ≤ f n x := by
+    intros n x hn
+    refine Nat.le_induction ?_ ?_ n hn
+    . rw [h₀ x]
+      exact NNReal.zero_le_coe
+    . intros d hd₀ hd₁
+      rw [h₁ d x hd₀]
+      refine mul_nonneg hd₁ ?_
+      refine add_nonneg hd₁ ?_
+      refine div_nonneg (by linarith) ?_
+      exact Nat.cast_nonneg' d
+  have hmo₀: ∀ n, 0 < n → StrictMono (f n) := by
+    intros n hn₀
+    refine Monotone.strictMono_of_injective ?h₁ ?h₂
+    . refine monotone_iff_forall_lt.mpr ?h₁.a
+      intros a b hab
+      refine le_of_lt ?_
+      exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n a b hn₀ hab
+    . intros p q hpq
+      contrapose! hpq
+      apply lt_or_gt_of_ne at hpq
+      cases' hpq with hpq hpq
+      . refine ne_of_lt ?_
+        exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n p q hn₀ hpq
+      . symm
+        refine ne_of_lt ?_
+        exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n q p hn₀ hpq
+  have hmo₁: ∀ n, 0 < n → Function.Injective (f n) := by exact fun n a => StrictMono.injective (hmo₀ n a)
+  let f₀: ℕ → NNReal → NNReal := fun n x => (f n x).toNNReal
+  have hf₀: f₀ = fun n x => (f n x).toNNReal := by rfl
+  have hf₁: ∀ n x, 0 < n → f n x = f₀ n x := by
+    intros n x hn₀
+    rw [hf₀]
+    simp
+    exact h₃ n x hn₀
+  have hf₂: ∀ n x, 0 < n → f₀ n x = (f n x).toNNReal := by
+    intros n x _
+    rw [hf₀]
+  have hmo₂: ∀ n, 0 < n → StrictMono (f₀ n) := by
+    intros n hn₀
+    refine imo_1985_p6_4 f h₃ ?_ f₀ hf₀ n hn₀
+    exact fun n x y a a_1 => hmo₀ n a a_1
+  let fi : ℕ → NNReal → NNReal := fun n => Function.invFun (f₀ n)
+  have hmo₇: ∀ n, 0 < n → Function.RightInverse (fi n) (f₀ n) := by
+    intros n hn₀
+    refine Function.rightInverse_invFun ?_
+    have h₄: ∀ n x y, 0 < n → x < y → f n x < f n y := by
+      exact fun n x y a a_1 => imo_1985_p6_2 f h₀ h₁ h₂ h₃ n x y a a_1
+    refine imo_1985_p6_7 f h₀ h₁ h₃ ?_ f₀ hf₂ hmo₂ ?_ n hn₀
+    . exact fun n x a => imo_1985_p6_3 f h₀ h₁ h₄ n x a
+    . intros m hm₀
+      exact imo_1985_p6_6 f h₀ h₁ f₀ hf₀ m hm₀
+  have hf₇: ∀ n x y, 0 < n → (f₀ n x = y ↔ fi n y = x) := by
+    intros n x y hn₀
+    constructor
+    . intro hn₁
+      rw [← hn₁, hf₀]
+      have hn₂: (Function.invFun (f n)) ∘ (f n) = id := by exact Function.invFun_comp (hmo₁ n hn₀)
+      rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+      exact imo_1985_p6_5 f hmo₁ f₀ hmo₂ fi rfl n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+    . intro hn₁
+      rw [← hn₁]
+      exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ y))
+  let sn : Set ℕ := Set.Ici 1
+  let fb : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n (1 - 1 / (n:NNReal)))
+  let fc : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n 1)
+  have hsn₁: ∀ n:↑sn, ↑n ∈ sn ∧ 0 < (↑n:ℕ) := by
+    intro n
+    have hn₀: ↑n ∈ sn := by exact Subtype.coe_prop n
+    constructor
+    . exact Subtype.coe_prop n
+    . exact hn₀
+  have hfb₀: fb = fun (n:↑sn) => fi n (1 - 1 / (n:NNReal)) := by rfl
+  have hfc₀: fc = fun (n:↑sn) => fi n 1 := by rfl
+  have hfb₁: ∀ n:↑sn, f₀ n (fb n) = 1 - 1 / (n:NNReal) := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfb₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ (1 - 1 / (n:NNReal))))
+  have hfc₁: ∀ n:↑sn, f₀ n (fc n) = 1 := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfc₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ 1))
+  have hu₁: ∀ n:↑sn, fb n < 1 := by
+    exact imo_1985_p6_8 f h₀ h₁ hmo₀ hmo₁ f₀ hf₂ sn fb hsn₁ hfb₁
+  have hfc₂: ∀ n:↑sn, fb n < fc n := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    have g₀: f₀ n (fb n) < f₀ n (fc n) := by
+      rw [hfb₁ n, hfc₁ n]
+      simp
+      exact (hsn₁ n).2
+    exact (StrictMono.lt_iff_lt (hmo₂ n hn₀)).mp g₀
+  have hfb₃: StrictMono fb := by
+    refine StrictMonoOn.restrict ?_
+    refine imo_1985_p6_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn (by rfl)
+    intro x
+    refine (hf₇ 1 x x (by linarith)).mp ?_
+    rw [hf₂ 1 x (by linarith), h₀]
+    exact Real.toNNReal_coe
+  have hfc₃: StrictAnti fc := by
+    have g₀: StrictAntiOn (fun n => fi n 1) sn := by
+      refine strictAntiOn_Ici_of_lt_pred ?_
+      intros m hm₀
+      have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+      have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+      have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+      simp
+      let x := fi m 1
+      let y := fi (m - 1) 1
+      have hx₀: x = fi m 1 := by rfl
+      have hy₀: y = fi (m - 1) 1 := by rfl
+      have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+      have hy₁: f₀ (m - 1) y = 1 := by
+        exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+      have hy₂: f (m - 1) y = 1 := by
+        rw [hf₁ (m - 1) y hm₁, hy₁]
+        exact rfl
+      have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+      refine (StrictMono.lt_iff_lt hf).mp ?_
+      rw [← hx₀, ← hy₀]
+      rw [hf₁ m x hm₃, hf₁ m y hm₃]
+      refine NNReal.coe_lt_coe.mpr ?_
+      rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+      simp
+      exact hm₀
+    intros m n hmn
+    rw [hfc₀]
+    simp
+    let mn : ℕ := ↑m
+    let nn : ℕ := ↑n
+    have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+    have hn₀: nn ��� sn := by exact Subtype.coe_prop n
+    exact g₀ hm₀ hn₀ hmn
+  let sb := Set.range fb
+  let sc := Set.range fc
+  have hsb₀: sb = Set.range fb := by rfl
+  have hsc₀: sc = Set.range fc := by rfl
+  let fr : NNReal → ℝ := fun x => x.toReal
+  let sbr := Set.image fr sb
+  let scr := Set.image fr sc
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . exact Set.Nonempty.of_subtype
+    . refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . refine Set.Nonempty.image fr ?_
+      exact Set.range_nonempty fc
+    . exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb rfl hfb₀ hsb₀ fr rfl sbr rfl br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by exact hfb₀
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn (by rfl) fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+
+lemma imo_1985_p6_main_2
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (n : ℕ)
+  (x : NNReal)
+  (hn : 0 < n):
+   0 ≤ f n x := by
+  refine Nat.le_induction ?_ ?_ n hn
+  . rw [h₀ x]
+    exact NNReal.zero_le_coe
+  . intros d hd₀ hd₁
+    rw [h₁ d x hd₀]
+    refine mul_nonneg hd₁ ?_
+    refine add_nonneg hd₁ ?_
+    refine div_nonneg (by linarith) ?_
+    exact Nat.cast_nonneg' d
+
+lemma imo_1985_p6_main_3
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (x : NNReal):
+   ∀ (n : ℕ), Nat.succ 0 ≤ n → 0 ≤ f n x → 0 ≤ f (n + 1) x := by
+  intros d hd₀ hd₁
+  rw [h₁ d x hd₀]
+  refine mul_nonneg hd₁ ?_
+  refine add_nonneg hd₁ ?_
+  refine div_nonneg (by linarith) ?_
+  exact Nat.cast_nonneg' d
+
+
+lemma imo_1985_p6_main_4
+  (f : ℕ → NNReal → ℝ)
+  (x : NNReal)
+  (d : ℕ)
+  (hd₁ : 0 ≤ f d x):
+   0 ≤ f d x * (f d x + 1 / ↑d) := by
+  refine mul_nonneg hd₁ ?_
+  refine add_nonneg hd₁ ?_
+  refine div_nonneg (by linarith) ?_
+  exact Nat.cast_nonneg' d
+
+
+lemma imo_1985_p6_main_5
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hmo₀: ∀ n, 0 < n → StrictMono (f n) := by
+    intros n hn₀
+    refine Monotone.strictMono_of_injective ?h₁ ?h₂
+    . refine monotone_iff_forall_lt.mpr ?h₁.a
+      intros a b hab
+      refine le_of_lt ?_
+      exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n a b hn₀ hab
+    . intros p q hpq
+      contrapose! hpq
+      apply lt_or_gt_of_ne at hpq
+      cases' hpq with hpq hpq
+      . refine ne_of_lt ?_
+        exact imo_1985_p6_2 f h₀ h₁ h₂ h�� n p q hn₀ hpq
+      . symm
+        refine ne_of_lt ?_
+        exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n q p hn₀ hpq
+  have hmo₁: ∀ n, 0 < n → Function.Injective (f n) := by exact fun n a => StrictMono.injective (hmo₀ n a)
+  let f₀: ℕ → NNReal → NNReal := fun n x => (f n x).toNNReal
+  have hf₀: f₀ = fun n x => (f n x).toNNReal := by rfl
+  have hf₁: ∀ n x, 0 < n → f n x = f₀ n x := by
+    intros n x hn₀
+    rw [hf₀]
+    simp
+    exact h₃ n x hn₀
+  have hf₂: ∀ n x, 0 < n → f₀ n x = (f n x).toNNReal := by
+    intros n x _
+    rw [hf₀]
+  have hmo₂: ∀ n, 0 < n → StrictMono (f₀ n) := by
+    intros n hn₀
+    refine imo_1985_p6_4 f h₃ ?_ f₀ hf₀ n hn₀
+    exact fun n x y a a_1 => hmo₀ n a a_1
+  let fi : ℕ → NNReal → NNReal := fun n => Function.invFun (f₀ n)
+  have hmo₇: ∀ n, 0 < n → Function.RightInverse (fi n) (f₀ n) := by
+    intros n hn₀
+    refine Function.rightInverse_invFun ?_
+    have h₄: ∀ n x y, 0 < n → x < y → f n x < f n y := by
+      exact fun n x y a a_1 => imo_1985_p6_2 f h₀ h₁ h₂ h₃ n x y a a_1
+    refine imo_1985_p6_7 f h₀ h₁ h₃ ?_ f₀ hf₂ hmo₂ ?_ n hn₀
+    . exact fun n x a => imo_1985_p6_3 f h₀ h₁ h₄ n x a
+    . intros m hm₀
+      exact imo_1985_p6_6 f h₀ h₁ f₀ hf₀ m hm₀
+  have hf₇: ∀ n x y, 0 < n → (f₀ n x = y ↔ fi n y = x) := by
+    intros n x y hn₀
+    constructor
+    . intro hn₁
+      rw [← hn₁, hf₀]
+      have hn₂: (Function.invFun (f n)) ∘ (f n) = id := by exact Function.invFun_comp (hmo₁ n hn₀)
+      rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+      exact imo_1985_p6_5 f hmo₁ f₀ hmo₂ fi rfl n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+    . intro hn₁
+      rw [← hn₁]
+      exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ y))
+  let sn : Set ℕ := Set.Ici 1
+  let fb : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n (1 - 1 / (n:NNReal)))
+  let fc : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n 1)
+  have hsn₁: ∀ n:↑sn, ↑n ∈ sn ∧ 0 < (↑n:ℕ) := by
+    intro n
+    have hn₀: ↑n ∈ sn := by exact Subtype.coe_prop n
+    constructor
+    . exact Subtype.coe_prop n
+    . exact hn₀
+  have hfb₀: fb = fun (n:↑sn) => fi n (1 - 1 / (n:NNReal)) := by rfl
+  have hfc₀: fc = fun (n:↑sn) => fi n 1 := by rfl
+  have hfb₁: ∀ n:↑sn, f₀ n (fb n) = 1 - 1 / (n:NNReal) := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfb₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ (1 - 1 / (n:NNReal))))
+  have hfc₁: ∀ n:↑sn, f₀ n (fc n) = 1 := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfc₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ 1))
+  have hu₁: ∀ n:↑sn, fb n < 1 := by
+    exact imo_1985_p6_8 f h₀ h₁ hmo₀ hmo₁ f₀ hf₂ sn fb hsn₁ hfb₁
+  have hfc₂: ∀ n:↑sn, fb n < fc n := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    have g₀: f₀ n (fb n) < f₀ n (fc n) := by
+      rw [hfb₁ n, hfc₁ n]
+      simp
+      exact (hsn₁ n).2
+    exact (StrictMono.lt_iff_lt (hmo₂ n hn₀)).mp g₀
+  have hfb₃: StrictMono fb := by
+    refine StrictMonoOn.restrict ?_
+    refine imo_1985_p6_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn (by rfl)
+    intro x
+    refine (hf₇ 1 x x (by linarith)).mp ?_
+    rw [hf₂ 1 x (by linarith), h₀]
+    exact Real.toNNReal_coe
+  have hfc₃: StrictAnti fc := by
+    have g₀: StrictAntiOn (fun n => fi n 1) sn := by
+      refine strictAntiOn_Ici_of_lt_pred ?_
+      intros m hm₀
+      have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+      have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+      have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+      simp
+      let x := fi m 1
+      let y := fi (m - 1) 1
+      have hx₀: x = fi m 1 := by rfl
+      have hy₀: y = fi (m - 1) 1 := by rfl
+      have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+      have hy₁: f₀ (m - 1) y = 1 := by
+        exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+      have hy₂: f (m - 1) y = 1 := by
+        rw [hf₁ (m - 1) y hm₁, hy₁]
+        exact rfl
+      have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+      refine (StrictMono.lt_iff_lt hf).mp ?_
+      rw [← hx₀, ← hy₀]
+      rw [hf₁ m x hm₃, hf₁ m y hm₃]
+      refine NNReal.coe_lt_coe.mpr ?_
+      rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+      simp
+      exact hm₀
+    intros m n hmn
+    rw [hfc₀]
+    simp
+    let mn : ℕ := ↑m
+    let nn : ℕ := ↑n
+    have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+    have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+    exact g₀ hm₀ hn₀ hmn
+  let sb := Set.range fb
+  let sc := Set.range fc
+  have hsb₀: sb = Set.range fb := by rfl
+  have hsc₀: sc = Set.range fc := by rfl
+  let fr : NNReal → ℝ := fun x => x.toReal
+  let sbr := Set.image fr sb
+  let scr := Set.image fr sc
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . exact Set.Nonempty.of_subtype
+    . refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . refine Set.Nonempty.image fr ?_
+      exact Set.range_nonempty fc
+    . exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb rfl hfb₀ hsb₀ fr rfl sbr rfl br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by exact hfb₀
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn (by rfl) fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+
+lemma imo_1985_p6_main_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   StrictMono (f n) := by
+  refine Monotone.strictMono_of_injective ?h₁ ?h₂
+  . refine monotone_iff_forall_lt.mpr ?h₁.a
+    intros a b hab
+    refine le_of_lt ?_
+    exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n a b hn₀ hab
+  . intros p q hpq
+    contrapose! hpq
+    apply lt_or_gt_of_ne at hpq
+    cases' hpq with hpq hpq
+    . refine ne_of_lt ?_
+      exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n p q hn₀ hpq
+    . symm
+      refine ne_of_lt ?_
+      exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n q p hn₀ hpq
+
+lemma imo_1985_p6_main_7
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Monotone (f n) := by
+  refine monotone_iff_forall_lt.mpr ?h₁.a
+  intros a b hab
+  refine le_of_lt ?_
+  exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n a b hn₀ hab
+
+lemma imo_1985_p6_main_8
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Function.Injective (f n) := by
+  intros p q hpq
+  contrapose! hpq
+  apply lt_or_gt_of_ne at hpq
+  cases' hpq with hpq hpq
+  . refine ne_of_lt ?_
+    exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n p q hn₀ hpq
+  . symm
+    refine ne_of_lt ?_
+    exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n q p hn₀ hpq
+
+lemma imo_1985_p6_main_9
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (p q : NNReal)
+  (hpq : p ≠ q):
+   f n p ≠ f n q := by
+  apply lt_or_gt_of_ne at hpq
+  cases' hpq with hpq hpq
+  . refine ne_of_lt ?_
+    exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n p q hn₀ hpq
+  . symm
+    refine ne_of_lt ?_
+    exact imo_1985_p6_2 f h₀ h₁ h₂ h₃ n q p hn₀ hpq
+
+lemma imo_1985_p6_main_10
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n)):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hmo₁: ∀ n, 0 < n → Function.Injective (f n) := by exact fun n a => StrictMono.injective (hmo₀ n a)
+  let f₀: ℕ → NNReal → NNReal := fun n x => (f n x).toNNReal
+  have hf₀: f₀ = fun n x => (f n x).toNNReal := by rfl
+  have hf₁: ∀ n x, 0 < n → f n x = f₀ n x := by
+    intros n x hn₀
+    rw [hf₀]
+    simp
+    exact h₃ n x hn₀
+  have hf₂: ∀ n x, 0 < n → f₀ n x = (f n x).toNNReal := by
+    intros n x _
+    rw [hf₀]
+  have hmo₂: ∀ n, 0 < n → StrictMono (f₀ n) := by
+    intros n hn₀
+    refine imo_1985_p6_4 f h₃ ?_ f₀ hf₀ n hn₀
+    exact fun n x y a a_1 => hmo₀ n a a_1
+  let fi : ℕ → NNReal → NNReal := fun n => Function.invFun (f₀ n)
+  have hmo₇: ∀ n, 0 < n → Function.RightInverse (fi n) (f₀ n) := by
+    intros n hn₀
+    refine Function.rightInverse_invFun ?_
+    have h₄: ∀ n x y, 0 < n → x < y → f n x < f n y := by
+      exact fun n x y a a_1 => imo_1985_p6_2 f h₀ h₁ h₂ h₃ n x y a a_1
+    refine imo_1985_p6_7 f h₀ h₁ h₃ ?_ f₀ hf₂ hmo₂ ?_ n hn₀
+    . exact fun n x a => imo_1985_p6_3 f h₀ h₁ h₄ n x a
+    . intros m hm₀
+      exact imo_1985_p6_6 f h₀ h₁ f₀ hf₀ m hm₀
+  have hf₇: ∀ n x y, 0 < n → (f₀ n x = y ↔ fi n y = x) := by
+    intros n x y hn₀
+    constructor
+    . intro hn₁
+      rw [← hn₁, hf₀]
+      have hn₂: (Function.invFun (f n)) ∘ (f n) = id := by exact Function.invFun_comp (hmo₁ n hn₀)
+      rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+      exact imo_1985_p6_5 f hmo₁ f₀ hmo₂ fi rfl n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+    . intro hn₁
+      rw [← hn₁]
+      exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ y))
+  let sn : Set ℕ := Set.Ici 1
+  let fb : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n (1 - 1 / (n:NNReal)))
+  let fc : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n 1)
+  have hsn₁: ∀ n:↑sn, ↑n ∈ sn ∧ 0 < (↑n:ℕ) := by
+    intro n
+    have hn₀: ↑n ∈ sn := by exact Subtype.coe_prop n
+    constructor
+    . exact Subtype.coe_prop n
+    . exact hn₀
+  have hfb₀: fb = fun (n:↑sn) => fi n (1 - 1 / (n:NNReal)) := by rfl
+  have hfc₀: fc = fun (n:↑sn) => fi n 1 := by rfl
+  have hfb₁: ∀ n:↑sn, f₀ n (fb n) = 1 - 1 / (n:NNReal) := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfb₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ (1 - 1 / (n:NNReal))))
+  have hfc₁: ∀ n:↑sn, f₀ n (fc n) = 1 := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfc₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ 1))
+  have hu₁: ∀ n:↑sn, fb n < 1 := by
+    exact imo_1985_p6_8 f h₀ h₁ hmo₀ hmo₁ f₀ hf₂ sn fb hsn₁ hfb₁
+  have hfc₂: ∀ n:↑sn, fb n < fc n := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    have g₀: f₀ n (fb n) < f₀ n (fc n) := by
+      rw [hfb₁ n, hfc₁ n]
+      simp
+      exact (hsn₁ n).2
+    exact (StrictMono.lt_iff_lt (hmo₂ n hn₀)).mp g₀
+  have hfb₃: StrictMono fb := by
+    refine StrictMonoOn.restrict ?_
+    refine imo_1985_p6_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn (by rfl)
+    intro x
+    refine (hf₇ 1 x x (by linarith)).mp ?_
+    rw [hf₂ 1 x (by linarith), h₀]
+    exact Real.toNNReal_coe
+  have hfc₃: StrictAnti fc := by
+    have g₀: StrictAntiOn (fun n => fi n 1) sn := by
+      refine strictAntiOn_Ici_of_lt_pred ?_
+      intros m hm₀
+      have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+      have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+      have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+      simp
+      let x := fi m 1
+      let y := fi (m - 1) 1
+      have hx₀: x = fi m 1 := by rfl
+      have hy₀: y = fi (m - 1) 1 := by rfl
+      have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+      have hy₁: f₀ (m - 1) y = 1 := by
+        exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+      have hy₂: f (m - 1) y = 1 := by
+        rw [hf₁ (m - 1) y hm₁, hy₁]
+        exact rfl
+      have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+      refine (StrictMono.lt_iff_lt hf).mp ?_
+      rw [← hx₀, ← hy₀]
+      rw [hf₁ m x hm₃, hf₁ m y hm₃]
+      refine NNReal.coe_lt_coe.mpr ?_
+      rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+      simp
+      exact hm₀
+    intros m n hmn
+    rw [hfc₀]
+    simp
+    let mn : ℕ := ↑m
+    let nn : ℕ := ↑n
+    have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+    have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+    exact g₀ hm₀ hn₀ hmn
+  let sb := Set.range fb
+  let sc := Set.range fc
+  have hsb₀: sb = Set.range fb := by rfl
+  have hsc₀: sc = Set.range fc := by rfl
+  let fr : NNReal → ℝ := fun x => x.toReal
+  let sbr := Set.image fr sb
+  let scr := Set.image fr sc
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . exact Set.Nonempty.of_subtype
+    . refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . refine Set.Nonempty.image fr ?_
+      exact Set.range_nonempty fc
+    . exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb rfl hfb₀ hsb₀ fr rfl sbr rfl br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by exact hfb₀
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn (by rfl) fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+
+lemma imo_1985_p6_main_11
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal):
+   ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x) := by
+  intros n x hn₀
+  rw [hf₀]
+  simp
+  exact h₃ n x hn₀
+
+lemma imo_1985_p6_main_12
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x)):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hf₂: ∀ n x, 0 < n → f₀ n x = (f n x).toNNReal := by
+    intros n x _
+    rw [hf₀]
+  have hmo₂: ∀ n, 0 < n → StrictMono (f₀ n) := by
+    intros n hn₀
+    refine imo_1985_p6_4 f h₃ ?_ f₀ hf₀ n hn₀
+    exact fun n x y a a_1 => hmo₀ n a a_1
+  let fi : ℕ → NNReal → NNReal := fun n => Function.invFun (f₀ n)
+  have hmo₇: ∀ n, 0 < n → Function.RightInverse (fi n) (f₀ n) := by
+    intros n hn₀
+    refine Function.rightInverse_invFun ?_
+    have h₄: ∀ n x y, 0 < n → x < y → f n x < f n y := by
+      exact fun n x y a a_1 => imo_1985_p6_2 f h₀ h₁ h₂ h₃ n x y a a_1
+    refine imo_1985_p6_7 f h₀ h₁ h₃ ?_ f₀ hf₂ hmo₂ ?_ n hn₀
+    . exact fun n x a => imo_1985_p6_3 f h₀ h₁ h₄ n x a
+    . intros m hm₀
+      exact imo_1985_p6_6 f h₀ h₁ f₀ hf₀ m hm₀
+  have hf₇: ∀ n x y, 0 < n → (f₀ n x = y ↔ fi n y = x) := by
+    intros n x y hn₀
+    constructor
+    . intro hn₁
+      rw [← hn₁, hf₀]
+      have hn₂: (Function.invFun (f n)) ∘ (f n) = id := by exact Function.invFun_comp (hmo₁ n hn₀)
+      rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+      exact imo_1985_p6_5 f hmo₁ f₀ hmo₂ fi rfl n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+    . intro hn₁
+      rw [← hn₁]
+      exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ y))
+  let sn : Set ℕ := Set.Ici 1
+  let fb : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n (1 - 1 / (n:NNReal)))
+  let fc : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n 1)
+  have hsn₁: ∀ n:↑sn, ↑n ∈ sn ∧ 0 < (↑n:ℕ) := by
+    intro n
+    have hn₀: ↑n ∈ sn := by exact Subtype.coe_prop n
+    constructor
+    . exact Subtype.coe_prop n
+    . exact hn₀
+  have hfb₀: fb = fun (n:↑sn) => fi n (1 - 1 / (n:NNReal)) := by rfl
+  have hfc₀: fc = fun (n:↑sn) => fi n 1 := by rfl
+  have hfb₁: ∀ n:↑sn, f₀ n (fb n) = 1 - 1 / (n:NNReal) := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfb₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ (1 - 1 / (n:NNReal))))
+  have hfc₁: ∀ n:↑sn, f₀ n (fc n) = 1 := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfc₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ 1))
+  have hu₁: ∀ n:↑sn, fb n < 1 := by
+    exact imo_1985_p6_8 f h₀ h₁ hmo₀ hmo₁ f₀ hf₂ sn fb hsn₁ hfb₁
+  have hfc₂: ∀ n:↑sn, fb n < fc n := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    have g₀: f₀ n (fb n) < f₀ n (fc n) := by
+      rw [hfb₁ n, hfc₁ n]
+      simp
+      exact (hsn₁ n).2
+    exact (StrictMono.lt_iff_lt (hmo₂ n hn₀)).mp g₀
+  have hfb₃: StrictMono fb := by
+    refine StrictMonoOn.restrict ?_
+    refine imo_1985_p6_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn (by rfl)
+    intro x
+    refine (hf₇ 1 x x (by linarith)).mp ?_
+    rw [hf₂ 1 x (by linarith), h₀]
+    exact Real.toNNReal_coe
+  have hfc₃: StrictAnti fc := by
+    have g₀: StrictAntiOn (fun n => fi n 1) sn := by
+      refine strictAntiOn_Ici_of_lt_pred ?_
+      intros m hm₀
+      have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+      have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+      have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+      simp
+      let x := fi m 1
+      let y := fi (m - 1) 1
+      have hx₀: x = fi m 1 := by rfl
+      have hy₀: y = fi (m - 1) 1 := by rfl
+      have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+      have hy₁: f₀ (m - 1) y = 1 := by
+        exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+      have hy₂: f (m - 1) y = 1 := by
+        rw [hf₁ (m - 1) y hm₁, hy₁]
+        exact rfl
+      have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+      refine (StrictMono.lt_iff_lt hf).mp ?_
+      rw [← hx₀, ← hy₀]
+      rw [hf₁ m x hm₃, hf₁ m y hm₃]
+      refine NNReal.coe_lt_coe.mpr ?_
+      rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+      simp
+      exact hm₀
+    intros m n hmn
+    rw [hfc₀]
+    simp
+    let mn : ℕ := ↑m
+    let nn : ℕ := ↑n
+    have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+    have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+    exact g₀ hm₀ hn₀ hmn
+  let sb := Set.range fb
+  let sc := Set.range fc
+  have hsb₀: sb = Set.range fb := by rfl
+  have hsc₀: sc = Set.range fc := by rfl
+  let fr : NNReal → ℝ := fun x => x.toReal
+  let sbr := Set.image fr sb
+  let scr := Set.image fr sc
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . exact Set.Nonempty.of_subtype
+    . refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . refine Set.Nonempty.image fr ?_
+      exact Set.range_nonempty fc
+    . exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb rfl hfb₀ hsb₀ fr rfl sbr rfl br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by exact hfb₀
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn (by rfl) fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+lemma imo_1985_p6_main_13
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   Function.RightInverse (fi n) (f₀ n) := by
+  rw [hfi]
+  refine Function.rightInverse_invFun ?_
+  have h₄: ∀ n x y, 0 < n → x < y → f n x < f n y := by
+    exact fun n x y a a_1 => imo_1985_p6_2 f h₀ h₁ h₂ h₃ n x y a a_1
+  refine imo_1985_p6_7 f h₀ h₁ h₃ ?_ f₀ hf₂ hmo₂ ?_ n hn₀
+  . exact fun n x a => imo_1985_p6_3 f h₀ h₁ h₄ n x a
+  . intros m hm₀
+    exact imo_1985_p6_6 f h₀ h₁ f₀ hf₀ m hm₀
+
+lemma imo_1985_p6_main_14
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y):
+   Function.Surjective (f₀ n) := by
+  refine imo_1985_p6_7 f h₀ h₁ h₃ ?_ f₀ hf₂ hmo₂ ?_ n hn₀
+  . exact fun n x a => imo_1985_p6_3 f h₀ h₁ h₄ n x a
+  . intros m hm₀
+    exact imo_1985_p6_6 f h₀ h₁ f₀ hf₀ m hm₀
+
+lemma imo_1985_p6_main_15
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n)):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hf₇: ∀ n x y, 0 < n → (f₀ n x = y ↔ fi n y = x) := by
+    intros n x y hn₀
+    constructor
+    . intro hn₁
+      rw [← hn₁, hf₀]
+      have hn₂: (Function.invFun (f n)) ∘ (f n) = id := by exact Function.invFun_comp (hmo₁ n hn₀)
+      rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+      exact imo_1985_p6_5 f hmo₁ f₀ hmo₂ fi hfi n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+    . intro hn₁
+      rw [← hn₁]
+      exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ y))
+  let sn : Set ℕ := Set.Ici 1
+  let fb : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n (1 - 1 / (n:NNReal)))
+  let fc : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n 1)
+  have hsn₁: ∀ n:↑sn, ↑n ∈ sn ∧ 0 < (↑n:ℕ) := by
+    intro n
+    have hn₀: ↑n ∈ sn := by exact Subtype.coe_prop n
+    constructor
+    . exact Subtype.coe_prop n
+    . exact hn₀
+  have hfb₀: fb = fun (n:↑sn) => fi n (1 - 1 / (n:NNReal)) := by rfl
+  have hfc₀: fc = fun (n:↑sn) => fi n 1 := by rfl
+  have hfb₁: ∀ n:↑sn, f₀ n (fb n) = 1 - 1 / (n:NNReal) := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfb₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ (1 - 1 / (n:NNReal))))
+  have hfc₁: ∀ n:↑sn, f₀ n (fc n) = 1 := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfc₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ 1))
+  have hu₁: ∀ n:↑sn, fb n < 1 := by
+    exact imo_1985_p6_8 f h₀ h₁ hmo₀ hmo₁ f₀ hf₂ sn fb hsn₁ hfb₁
+  have hfc₂: ∀ n:↑sn, fb n < fc n := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    have g₀: f₀ n (fb n) < f₀ n (fc n) := by
+      rw [hfb₁ n, hfc₁ n]
+      simp
+      exact (hsn₁ n).2
+    exact (StrictMono.lt_iff_lt (hmo₂ n hn₀)).mp g₀
+  have hfb₃: StrictMono fb := by
+    refine StrictMonoOn.restrict ?_
+    refine imo_1985_p6_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn (by rfl)
+    intro x
+    refine (hf₇ 1 x x (by linarith)).mp ?_
+    rw [hf₂ 1 x (by linarith), h₀]
+    exact Real.toNNReal_coe
+  have hfc₃: StrictAnti fc := by
+    have g₀: StrictAntiOn (fun n => fi n 1) sn := by
+      refine strictAntiOn_Ici_of_lt_pred ?_
+      intros m hm₀
+      have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+      have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+      have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+      simp
+      let x := fi m 1
+      let y := fi (m - 1) 1
+      have hx₀: x = fi m 1 := by rfl
+      have hy₀: y = fi (m - 1) 1 := by rfl
+      have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+      have hy₁: f₀ (m - 1) y = 1 := by
+        exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+      have hy₂: f (m - 1) y = 1 := by
+        rw [hf₁ (m - 1) y hm₁, hy₁]
+        exact rfl
+      have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+      refine (StrictMono.lt_iff_lt hf).mp ?_
+      rw [← hx₀, ← hy₀]
+      rw [hf₁ m x hm₃, hf₁ m y hm₃]
+      refine NNReal.coe_lt_coe.mpr ?_
+      rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+      simp
+      exact hm₀
+    intros m n hmn
+    rw [hfc₀]
+    simp
+    let mn : ℕ := ↑m
+    let nn : ℕ := ↑n
+    have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+    have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+    exact g₀ hm₀ hn₀ hmn
+  let sb := Set.range fb
+  let sc := Set.range fc
+  have hsb₀: sb = Set.range fb := by rfl
+  have hsc₀: sc = Set.range fc := by rfl
+  let fr : NNReal → ℝ := fun x => x.toReal
+  let sbr := Set.image fr sb
+  let scr := Set.image fr sc
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . exact Set.Nonempty.of_subtype
+    . refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . refine Set.Nonempty.image fr ?_
+      exact Set.range_nonempty fc
+    . exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb rfl hfb₀ hsb₀ fr rfl sbr rfl br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by rw [hfb₀, hfi]
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn (by rfl) fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+lemma imo_1985_p6_main_16
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (n : ℕ)
+  (x y : NNReal)
+  (hn₀ : 0 < n):
+   f₀ n x = y ↔ fi n y = x := by
+  constructor
+  . intro hn₁
+    rw [← hn₁, hf₀]
+    have hn₂: (Function.invFun (f n)) ∘ (f n) = id := by exact Function.invFun_comp (hmo₁ n hn₀)
+    rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+    exact imo_1985_p6_5 f hmo₁ f₀ hmo₂ fi hfi n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+  . intro hn₁
+    rw [← hn₁]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ y))
+
+
+lemma imo_1985_p6_main_17
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x ↦ (f n x).toNNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (n : ℕ)
+  (x y : NNReal)
+  (hn₀ : 0 < n)
+  (hn₁ : f₀ n x = y):
+   fi n y = x := by
+  rw [← hn₁, hf₀]
+  have hn₂: (Function.invFun (f n)) ∘ (f n) = id := by exact Function.invFun_comp (hmo₁ n hn₀)
+  rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+  exact imo_1985_p6_5 f hmo₁ f₀ hmo₂ fi hfi n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+
+lemma imo_1985_p6_main_18
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 0 < n) :
+   fi n ((fun n x ↦ (f n x).toNNReal) n x) = x := by
+  have hn₂: (Function.invFun (f n)) ∘ (f n) = id := by exact Function.invFun_comp (hmo₁ n hn₀)
+  rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+  exact imo_1985_p6_5 f hmo₁ f₀ hmo₂ fi hfi n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+
+lemma imo_1985_p6_main_19
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 0 < n)
+  (hn₂ : Function.invFun (f n) ∘ f n = id):
+   fi n ((fun n x ↦ (f n x).toNNReal) n x) = x := by
+  rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+  exact imo_1985_p6_5 f hmo₁ f₀ hmo₂ fi hfi n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+
+lemma imo_1985_p6_main_20
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (n : ℕ)
+  (x y : NNReal)
+  (hn₀ : 0 < n):
+   fi n y = x → f₀ n x = y := by
+  intro hn₁
+  rw [← hn₁]
+  exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ y))
+
+lemma imo_1985_p6_main_21
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x ��� 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₀: fb = sn.restrict fun n => fi n (1 - 1 / (n:NNReal)))
+  (hfc₀ : fc = sn.restrict fun n ↦ fi n 1):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hsn₁: ∀ n:↑sn, ↑n ∈ sn ∧ 0 < (↑n:ℕ) := by
+    intro n
+    constructor
+    . exact Subtype.coe_prop n
+    . refine Nat.lt_of_succ_le ?_
+      refine Set.mem_Ici.mp ?_
+      rw [← hsn]
+      exact n.2
+  have hfb₁: ∀ n:↑sn, f₀ n (fb n) = 1 - 1 / (n:NNReal) := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfb₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ (1 - 1 / (n:NNReal))))
+  have hfc₁: ∀ n:↑sn, f₀ n (fc n) = 1 := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfc₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ 1))
+  have hu₁: ∀ n:↑sn, fb n < 1 := by
+    exact imo_1985_p6_8 f h₀ h₁ hmo₀ hmo₁ f₀ hf₂ sn fb hsn₁ hfb₁
+  have hfc₂: ∀ n:↑sn, fb n < fc n := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    have g₀: f₀ n (fb n) < f₀ n (fc n) := by
+      rw [hfb₁ n, hfc₁ n]
+      simp
+      exact (hsn₁ n).2
+    exact (StrictMono.lt_iff_lt (hmo₂ n hn₀)).mp g₀
+  have hfb₃: StrictMono fb := by
+    rw [hfb₀]
+    refine StrictMonoOn.restrict ?_
+    refine imo_1985_p6_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn hsn
+    intro x
+    refine (hf₇ 1 x x (by linarith)).mp ?_
+    rw [hf₂ 1 x (by linarith), h₀]
+    exact Real.toNNReal_coe
+  have hfc₃: StrictAnti fc := by
+    have g₀: StrictAntiOn (fun n => fi n 1) sn := by
+      rw [hsn]
+      refine strictAntiOn_Ici_of_lt_pred ?_
+      intros m hm₀
+      have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+      have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+      have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+      simp
+      let x := fi m 1
+      let y := fi (m - 1) 1
+      have hx₀: x = fi m 1 := by rfl
+      have hy₀: y = fi (m - 1) 1 := by rfl
+      have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+      have hy₁: f₀ (m - 1) y = 1 := by
+        exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+      have hy₂: f (m - 1) y = 1 := by
+        rw [hf₁ (m - 1) y hm₁, hy₁]
+        exact rfl
+      have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+      refine (StrictMono.lt_iff_lt hf).mp ?_
+      rw [← hx₀, ← hy₀]
+      rw [hf₁ m x hm₃, hf₁ m y hm₃]
+      refine NNReal.coe_lt_coe.mpr ?_
+      rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+      simp
+      exact hm₀
+    intros m n hmn
+    rw [hfc₀]
+    simp
+    let mn : ℕ := ↑m
+    let nn : ℕ := ↑n
+    have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+    have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+    exact g₀ hm₀ hn₀ hmn
+  let sb := Set.range fb
+  let sc := Set.range fc
+  have hsb₀: sb = Set.range fb := by rfl
+  have hsc₀: sc = Set.range fc := by rfl
+  let fr : NNReal → ℝ := fun x => x.toReal
+  let sbr := Set.image fr sb
+  let scr := Set.image fr sc
+  have hsn₂: Nonempty ↑sn := by
+    rw [hsn]
+    exact Set.nonempty_Ici_subtype
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . exact Set.Nonempty.of_subtype
+    . refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . refine Set.Nonempty.image fr ?_
+      rw [hsc₀]
+      exact Set.range_nonempty fc
+    . exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb hsn hfb₀ hsb₀ fr rfl sbr rfl br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by
+      rw [hfb₀, hfi]
+      exact rfl
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+lemma imo_1985_p6_main_22
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (n : ↑sn):
+  n.1 ∈ sn ∧ 0 < n.1 := by
+  constructor
+  . exact Subtype.coe_prop n
+  . refine Nat.lt_of_succ_le ?_
+    refine Set.mem_Ici.mp ?_
+    rw [← hsn]
+    exact n.2
+
+lemma imo_1985_p6_main_23
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hsn₁ : ∀ (n : ↑sn), n.1 ∈ sn ∧ 0 < n.1)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n)) :
+   ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n := by
+  intros n
+  have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+  rw [hfb₀]
+  exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ (1 - 1 / (n:NNReal))))
+
+
+lemma imo_1985_p6_main_24
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (hsn₁ : ∀ (n : ↑sn), n.1 ∈ sn ∧ 0 < n.1)
+  (hfc₀ : fc = sn.restrict fun n ↦ fi (↑n) 1)
+  (n : ↑sn):
+   f₀ (↑n) (fc n) = 1 := by
+  have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+  rw [hfc₀]
+  exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ 1))
+
+lemma imo_1985_p6_main_25
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hsn₁ : ∀ (n : ↑sn), n.1 ∈ sn ∧ 0 < n.1)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1) :
+   ∀ (n : ↑sn), fb n < fc n := by
+  intros n
+  have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+  have g₀: f₀ n (fb n) < f₀ n (fc n) := by
+    rw [hfb₁ n, hfc₁ n]
+    simp
+    exact (hsn₁ n).2
+  exact (StrictMono.lt_iff_lt (hmo₂ n hn₀)).mp g₀
+
+
+lemma imo_1985_p6_main_26
+  (f₀ : ℕ → NNReal → NNReal)
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hsn₁ : ∀ (n : ↑sn), n.1 ∈ sn ∧ 0 < n.1)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (n : ↑sn) :
+   f₀ (↑n) (fb n) < f₀ (↑n) (fc n) := by
+  rw [hfb₁ n, hfc₁ n]
+  simp
+  exact (hsn₁ n).2
+
+lemma imo_1985_p6_main_27
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (g₀ : f₀ (↑n) (fb n) < f₀ (↑n) (fc n)):
+   fb n < fc n := by
+  refine (StrictMono.lt_iff_lt ?_).mp g₀
+  exact hmo₂ n hn₀
+
+lemma imo_1985_p6_main_28
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n))
+  (hfc₀ : fc = sn.restrict fun n ↦ fi (↑n) 1)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hu₁ : ∀ (n : ↑sn), fb n < 1)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hfb₃: StrictMono fb := by
+    rw [hfb₀]
+    refine StrictMonoOn.restrict ?_
+    refine imo_1985_p6_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn hsn
+    intro x
+    refine (hf₇ 1 x x (by linarith)).mp ?_
+    rw [hf₂ 1 x (by linarith), h₀]
+    exact Real.toNNReal_coe
+  have hfc₃: StrictAnti fc := by
+    have g₀: StrictAntiOn (fun n => fi n 1) sn := by
+      rw [hsn]
+      refine strictAntiOn_Ici_of_lt_pred ?_
+      intros m hm₀
+      have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+      have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+      have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+      simp
+      let x := fi m 1
+      let y := fi (m - 1) 1
+      have hx₀: x = fi m 1 := by rfl
+      have hy₀: y = fi (m - 1) 1 := by rfl
+      have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+      have hy₁: f₀ (m - 1) y = 1 := by
+        exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+      have hy₂: f (m - 1) y = 1 := by
+        rw [hf₁ (m - 1) y hm₁, hy₁]
+        exact rfl
+      have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+      refine (StrictMono.lt_iff_lt hf).mp ?_
+      rw [← hx₀, ← hy₀]
+      rw [hf₁ m x hm₃, hf₁ m y hm₃]
+      refine NNReal.coe_lt_coe.mpr ?_
+      rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+      simp
+      exact hm₀
+    intros m n hmn
+    rw [hfc₀]
+    simp
+    let mn : ℕ := ↑m
+    let nn : ℕ := ↑n
+    have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+    have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+    exact g₀ hm₀ hn₀ hmn
+  let sb := Set.range fb
+  let sc := Set.range fc
+  have hsb₀: sb = Set.range fb := by rfl
+  have hsc₀: sc = Set.range fc := by rfl
+  let fr : NNReal → ℝ := fun x => x.toReal
+  let sbr := Set.image fr sb
+  let scr := Set.image fr sc
+  have hsn₂: Nonempty ↑sn := by
+    rw [hsn]
+    exact Set.nonempty_Ici_subtype
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . exact Set.Nonempty.of_subtype
+    . refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . refine Set.Nonempty.image fr ?_
+      exact Set.range_nonempty fc
+    . exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb hsn hfb₀ hsb₀ fr rfl sbr rfl br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by
+      rw [hfb₀, hfi]
+      exact rfl
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+lemma imo_1985_p6_main_29
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n)) :
+  StrictMono fb := by
+  rw [hfb₀]
+  refine StrictMonoOn.restrict ?_
+  refine imo_1985_p6_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn hsn
+  intro x
+  refine (hf₇ 1 x x (by linarith)).mp ?_
+  rw [hf₂ 1 x (by linarith), h₀]
+  exact Real.toNNReal_coe
+
+lemma imo_1985_p6_main_30
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1):
+  StrictMonoOn (fun n ↦ fi n (1 - 1 / ↑n)) sn := by
+  refine imo_1985_p6_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn hsn
+  intro x
+  refine (hf₇ 1 x x (by linarith)).mp ?_
+  rw [hf₂ 1 x (by linarith), h₀]
+  exact Real.toNNReal_coe
+
+lemma imo_1985_p6_main_31
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (x : NNReal):
+   fi 1 x = x := by
+  refine (hf₇ 1 x x (by linarith)).mp ?_
+  rw [hf₂ 1 x (by linarith), h₀]
+  exact Real.toNNReal_coe
+
+lemma imo_1985_p6_main_32
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₀ : fc = sn.restrict fun n ↦ fi (↑n) 1):
+   StrictAnti fc := by
+  have g₀: StrictAntiOn (fun n => fi n 1) sn := by
+    rw [hsn]
+    refine strictAntiOn_Ici_of_lt_pred ?_
+    intros m hm₀
+    have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+    have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+    have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+    simp
+    let x := fi m 1
+    let y := fi (m - 1) 1
+    have hx₀: x = fi m 1 := by rfl
+    have hy₀: y = fi (m - 1) 1 := by rfl
+    have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+    have hy₁: f₀ (m - 1) y = 1 := by
+      exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+    have hy₂: f (m - 1) y = 1 := by
+      rw [hf₁ (m - 1) y hm₁, hy₁]
+      exact rfl
+    have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+    refine (StrictMono.lt_iff_lt hf).mp ?_
+    rw [← hx₀, ← hy₀]
+    rw [hf₁ m x hm₃, hf₁ m y hm₃]
+    refine NNReal.coe_lt_coe.mpr ?_
+    rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+    simp
+    exact hm₀
+  intros m n hmn
+  rw [hfc₀]
+  simp
+  let mn : ℕ := ↑m
+  let nn : ℕ := ↑n
+  have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+  have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+  exact g₀ hm₀ hn₀ hmn
+
+lemma imo_1985_p6_main_33
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo�� : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1) :
+  StrictAntiOn (fun n ↦ fi n 1) sn := by
+  rw [hsn]
+  refine strictAntiOn_Ici_of_lt_pred ?_
+  intros m hm₀
+  have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+  have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+  have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+  simp
+  let x := fi m 1
+  let y := fi (m - 1) 1
+  have hx₀: x = fi m 1 := by rfl
+  have hy₀: y = fi (m - 1) 1 := by rfl
+  have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+  have hy₁: f₀ (m - 1) y = 1 := by
+    exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+  have hy₂: f (m - 1) y = 1 := by
+    rw [hf₁ (m - 1) y hm₁, hy₁]
+    exact rfl
+  have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+  refine (StrictMono.lt_iff_lt hf).mp ?_
+  rw [← hx₀, ← hy₀]
+  rw [hf₁ m x hm₃, hf₁ m y hm₃]
+  refine NNReal.coe_lt_coe.mpr ?_
+  rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+  simp
+  exact hm₀
+
+lemma imo_1985_p6_main_34
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (m : ℕ)
+  (hm₀ : 1 < m):
+   fi m 1 < fi (Order.pred m) 1 := by
+  have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+  have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+  have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+  simp
+  let x := fi m 1
+  let y := fi (m - 1) 1
+  have hx₀: x = fi m 1 := by rfl
+  have hy₀: y = fi (m - 1) 1 := by rfl
+  have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+  have hy₁: f₀ (m - 1) y = 1 := by
+    exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+  have hy₂: f (m - 1) y = 1 := by
+    rw [hf₁ (m - 1) y hm₁, hy₁]
+    exact rfl
+  have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+  refine (StrictMono.lt_iff_lt hf).mp ?_
+  rw [← hx₀, ← hy₀]
+  rw [hf₁ m x hm₃, hf₁ m y hm₃]
+  refine NNReal.coe_lt_coe.mpr ?_
+  rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+  simp
+  exact hm₀
+
+lemma imo_1985_p6_main_35
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (m : ℕ)
+  (hm₀ : 1 < m)
+  (hm₁ : 0 < m - 1)
+  (hm₂ : m = m - 1 + 1)
+  (hm₃ : 0 < m):
+   fi m 1 < fi (m - 1) 1 := by
+  let x := fi m 1
+  let y := fi (m - 1) 1
+  have hx₀: x = fi m 1 := by rfl
+  have hy₀: y = fi (m - 1) 1 := by rfl
+  have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+  have hy₁: f₀ (m - 1) y = 1 := by
+    exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+  have hy₂: f (m - 1) y = 1 := by
+    rw [hf₁ (m - 1) y hm₁, hy₁]
+    exact rfl
+  have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+  refine (StrictMono.lt_iff_lt hf).mp ?_
+  rw [← hx₀, ← hy₀]
+  rw [hf₁ m x hm₃, hf₁ m y hm₃]
+  refine NNReal.coe_lt_coe.mpr ?_
+  rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+  simp
+  exact hm₀
+
+lemma imo_1985_p6_main_36
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (m : ℕ)
+  (hm₀ : 1 < m)
+  (hm₁ : 0 < m - 1)
+  (hm₂ : m = m - 1 + 1)
+  (hm₃ : 0 < m)
+  (x : NNReal := fi m 1)
+  (y : NNReal := fi (m - 1) 1)
+  (hx₀ : x = fi m 1)
+  (hy₀ : y = fi (m - 1) 1)
+  (hx₁ : f₀ m x = 1)
+  (hy₂ : f (m - 1) y = 1):
+   fi m 1 < fi (m - 1) 1 := by
+  have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+  refine (StrictMono.lt_iff_lt hf).mp ?_
+  rw [← hx₀, ← hy₀]
+  rw [hf₁ m x hm₃, hf₁ m y hm₃]
+  refine NNReal.coe_lt_coe.mpr ?_
+  rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+  simp
+  exact hm₀
+
+lemma imo_1985_p6_main_37
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (m : ℕ)
+  (hm₀ : 1 < m)
+  (hm₁ : 0 < m - 1)
+  (hm₂ : m = m - 1 + 1)
+  (hm₃ : 0 < m)
+  (x : NNReal := fi m 1)
+  (y : NNReal := fi (m - 1) 1)
+  (hx₀ : x = fi m 1)
+  (hy₀ : y = fi (m - 1) 1)
+  (hx₁ : f₀ m x = 1)
+  (hy₂ : f (m - 1) y = 1)
+  (hf : StrictMono (f m)):
+   fi m 1 < fi (m - 1) 1 := by
+  refine (StrictMono.lt_iff_lt hf).mp ?_
+  rw [← hx₀, ← hy₀]
+  rw [hf₁ m x hm₃, hf₁ m y hm₃]
+  refine NNReal.coe_lt_coe.mpr ?_
+  rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+  simp
+  exact hm₀
+
+lemma imo_1985_p6_main_38
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (m : ℕ)
+  (hm₀ : 1 < m)
+  (hm₁ : 0 < m - 1)
+  (hm₂ : m = m - 1 + 1)
+  (hm₃ : 0 < m)
+  (x : NNReal := fi m 1)
+  (y : NNReal := fi (m - 1) 1)
+  (hx₀ : x = fi m 1)
+  (hy₀ : y = fi (m - 1) 1)
+  (hx₁ : f₀ m x = 1)
+  (hy₂ : f (m - 1) y = 1):
+   f m (fi m 1) < f m (fi (m - 1) 1) := by
+  rw [← hx₀, ← hy₀]
+  rw [hf₁ m x hm₃, hf₁ m y hm₃]
+  refine NNReal.coe_lt_coe.mpr ?_
+  rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+  simp
+  exact hm₀
+
+lemma imo_1985_p6_main_39
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (m : ℕ)
+  (hm₀ : 1 < m)
+  (hm₁ : 0 < m - 1)
+  (hm₂ : m = m - 1 + 1)
+  (hm₃ : 0 < m)
+  (x : NNReal := fi m 1)
+  (y : NNReal := fi (m - 1) 1)
+  (hx₁ : f₀ m x = 1)
+  (hy₂ : f (m - 1) y = 1):
+   f m x < f m y := by
+  rw [hf₁ m x hm₃, hf₁ m y hm₃]
+  refine NNReal.coe_lt_coe.mpr ?_
+  rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+  simp
+  exact hm₀
+
+lemma imo_1985_p6_main_40
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (m : ℕ)
+  (hm₀ : 1 < m)
+  (hm₁ : 0 < m - 1)
+  (hm₂ : m = m - 1 + 1)
+  (hm₃ : 0 < m)
+  (x : NNReal := fi m 1)
+  (y : NNReal := fi (m - 1) 1)
+  (hx₁ : f₀ m x = 1)
+  (hy₂ : f (m - 1) y = 1):
+  (↑(f₀ m x):ℝ) < ↑(f₀ m y) := by
+  refine NNReal.coe_lt_coe.mpr ?_
+  rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+  simp
+  exact hm₀
+
+lemma imo_1985_p6_main_41
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (m : ℕ)
+  (hm₀ : 1 < m)
+  (hm₁ : 0 < m - 1)
+  (hm₂ : m = m - 1 + 1)
+  (hm₃ : 0 < m)
+  (x : NNReal := fi m 1)
+  (y : NNReal := fi (m - 1) 1)
+  (hx₁ : f₀ m x = 1)
+  (hy₂ : f (m - 1) y = 1):
+   f₀ m x < f₀ m y := by
+  rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+  simp
+  exact hm₀
+
+lemma imo_1985_p6_main_42
+  (fi : ℕ → NNReal → NNReal)
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (hfc₀ : fc = sn.restrict fun n ↦ fi (↑n) 1)
+  (g₀ : StrictAntiOn (fun n ↦ fi n 1) sn)
+  (m n : ↑sn)
+  (hmn : m < n):
+   fc n < fc m := by
+  rw [hfc₀]
+  simp
+  let mn : ℕ := ↑m
+  let nn : ℕ := ↑n
+  have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+  have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+  exact g₀ hm₀ hn₀ hmn
+
+lemma imo_1985_p6_main_43
+  (fi : ℕ → NNReal → NNReal)
+  (sn : Set ℕ)
+  (g₀ : StrictAntiOn (fun n ↦ fi n 1) sn)
+  (m n : ↑sn)
+  (hmn : m < n):
+   fi (↑n) 1 < fi (↑m) 1 := by
+  let mn : ℕ := ↑m
+  let nn : ℕ := ↑n
+  have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+  have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+  exact g₀ hm₀ hn₀ hmn
+
+lemma imo_1985_p6_main_44
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n))
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hu₁ : ∀ (n : ↑sn), fb n < 1)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  let sb := Set.range fb
+  let sc := Set.range fc
+  have hsb₀: sb = Set.range fb := by rfl
+  have hsc₀: sc = Set.range fc := by rfl
+  let fr : NNReal → ℝ := fun x => x.toReal
+  let sbr := Set.image fr sb
+  let scr := Set.image fr sc
+  have hsn₂: Nonempty ↑sn := by
+    rw [hsn]
+    exact Set.nonempty_Ici_subtype
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . exact Set.Nonempty.of_subtype
+    . refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . refine Set.Nonempty.image fr ?_
+      exact Set.range_nonempty fc
+    . exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb hsn hfb₀ hsb₀ fr rfl sbr rfl br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by
+      rw [hfb₀, hfi]
+      exact rfl
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+
+lemma imo_1985_p6_main_45
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n))
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hu₁ : ∀ (n : ↑sn), fb n < 1)
+  (hfc₂ : �� (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hsn₂: Nonempty ↑sn := by
+    rw [hsn]
+    exact Set.nonempty_Ici_subtype
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . rw [hsbr]
+      refine Set.Nonempty.image fr ?_
+      rw [hsb₀]
+      exact Set.range_nonempty fb
+    . rw [hsbr, hfr]
+      refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        rw [hsb₀] at hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . rw [hscr]
+      refine Set.Nonempty.image fr ?_
+      rw [hsc₀]
+      exact Set.range_nonempty fc
+    . rw [hscr, hfr]
+      exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb hsn hfb₀ hsb₀ fr hfr sbr hsbr br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by
+      rw [hfb₀, hfi]
+      exact rfl
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+
+lemma imo_1985_p6_main_46
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hu₁ : ∀ (n : ↑sn), fb n < 1)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun (x:NNReal) ↦ (↑x:ℝ))
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb):
+   ∃ br, IsLUB sbr br := by
+  have hsn₂: Nonempty ↑sn := by
+    rw [hsn]
+    exact Set.nonempty_Ici_subtype
+  refine Real.exists_isLUB ?_ ?_
+  . rw [hsbr]
+    refine Set.Nonempty.image fr ?_
+    rw [hsb₀]
+    refine Set.range_nonempty fb
+  . rw [hsbr, hfr]
+    refine NNReal.bddAbove_coe.mpr ?_
+    refine (bddAbove_iff_exists_ge 1).mpr ?_
+    use 1
+    constructor
+    . exact Preorder.le_refl 1
+    . intros y hy₀
+      rw [hsb₀] at hy₀
+      apply Set.mem_range.mp at hy₀
+      obtain ⟨na, hna₀⟩ := hy₀
+      refine le_of_lt ?_
+      rw [← hna₀]
+      exact hu₁ na
+
+
+lemma imo_1985_p6_main_47
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hu₁ : ∀ (n : ↑sn), fb n < 1)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb) :
+  BddAbove sbr := by
+  rw [hsbr, hfr]
+  refine NNReal.bddAbove_coe.mpr ?_
+  refine (bddAbove_iff_exists_ge 1).mpr ?_
+  use 1
+  constructor
+  . exact Preorder.le_refl 1
+  . intros y hy₀
+    rw [hsb₀] at hy₀
+    apply Set.mem_range.mp at hy₀
+    obtain ⟨na, hna₀⟩ := hy₀
+    refine le_of_lt ?_
+    rw [← hna₀]
+    exact hu₁ na
+
+
+lemma imo_1985_p6_main_50
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hu₁ : ∀ (n : ↑sn), fb n < 1)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb):
+   ∀ y ∈ sb, y ≤ 1 := by
+  intros y hy���
+  rw [hsb₀] at hy₀
+  apply Set.mem_range.mp at hy₀
+  obtain ⟨na, hna₀⟩ := hy₀
+  refine le_of_lt ?_
+  rw [← hna₀]
+  exact hu₁ na
+
+lemma imo_1985_p6_main_51
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n))
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (hu₃ : ∃ br, IsLUB sbr br):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hsn₂: Nonempty ↑sn := by
+    rw [hsn]
+    exact Set.nonempty_Ici_subtype
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . rw [hscr]
+      refine Set.Nonempty.image fr ?_
+      rw [hsc₀]
+      exact Set.range_nonempty fc
+    . rw [hscr, hfr]
+      exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb hsn hfb₀ hsb₀ fr hfr sbr hsbr br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by
+      rw [hfb₀, hfi]
+      exact rfl
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+lemma imo_1985_p6_main_53
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n))
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (hu₃ : ∃ br, IsLUB sbr br)
+  (hu₄ : ∃ cr, IsGLB scr cr):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb hsn hfb₀ hsb₀ fr hfr sbr hsbr br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by
+      rw [hfb₀, hfi]
+      exact rfl
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+lemma imo_1985_p6_main_56
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n))
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (br : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (cr : ℝ)
+  (hcr₀ : IsGLB scr cr)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hbr₁: 0 < br := by
+    exact imo_1985_p6_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb hsn hfb₀ hsb₀ fr hfr sbr hsbr br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by
+      rw [hfb₀, hfi]
+      exact rfl
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+lemma imo_1985_p6_main_57
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n))
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (br : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (cr : ℝ)
+  (hcr₀ : IsGLB scr cr)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by
+      rw [hfb₀, hfi]
+      exact rfl
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+lemma imo_1985_p6_main_59
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (br : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (cr : ℝ)
+  (hcr₀ : IsGLB scr cr)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hu₅: br ≤ cr := by
+    exact imo_1985_p6_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+
+lemma imo_1985_p6_main_60
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (br : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (cr : ℝ)
+  (hcr₀ : IsGLB scr cr)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hu₅ : br ≤ cr):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+lemma imo_1985_p6_main_48
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hu₁ : ∀ (n : ↑sn), fb n < 1)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb):
+   BddAbove sb := by
+  refine (bddAbove_iff_exists_ge 1).mpr ?_
+  use 1
+  constructor
+  . exact Preorder.le_refl 1
+  . intros y hy₀
+    rw [hsb₀] at hy₀
+    apply Set.mem_range.mp at hy₀
+    obtain ⟨na, hna₀⟩ := hy₀
+    refine le_of_lt ?_
+    rw [← hna₀]
+    exact hu₁ na
+
+lemma imo_1985_p6_main_49
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hu₁ : ∀ (n : ↑sn), fb n < 1)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb):
+   ∃ x, 1 ≤ x ∧ ∀ y ∈ sb, y ≤ x := by
+  use 1
+  constructor
+  . exact Preorder.le_refl 1
+  . intros y hy₀
+    rw [hsb₀] at hy₀
+    apply Set.mem_range.mp at hy₀
+    obtain ⟨na, hna₀⟩ := hy₀
+    refine le_of_lt ?_
+    rw [← hna₀]
+    exact hu₁ na
+
+
+
+lemma imo_1985_p6_main_52
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc):
+   ∃ cr, IsGLB scr cr := by
+  have hsn₂: Nonempty ↑sn := by
+    rw [hsn]
+    exact Set.nonempty_Ici_subtype
+  refine Real.exists_isGLB ?_ ?_
+  . rw [hscr]
+    refine Set.Nonempty.image fr ?_
+    rw [hsc₀]
+    exact Set.range_nonempty fc
+  . rw [hscr, hfr]
+    exact NNReal.bddBelow_coe sc
+
+
+
+lemma imo_1985_p6_main_54
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 0 < n)
+  (hn₁ : f n x < f (n + 1) x):
+   1 - 1 / ↑n < f n x := by
+  rw [h₁ n x hn₀] at hn₁
+  nth_rw 1 [← mul_one (f n x)] at hn₁
+  suffices g₀: 1 < f n x + 1 / ↑n
+  . exact sub_right_lt_of_lt_add g₀
+  . refine lt_of_mul_lt_mul_left hn₁ ?_
+    exact h₃ n x hn₀
+
+lemma imo_1985_p6_main_55
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (n : ℕ)
+  (x : NNReal)
+  (hn₀ : 0 < n)
+  (hx₀ : 0 < x)
+  (hn₁ : 1 - 1 / ↑n < f n x):
+   f n x < f (n + 1) x := by
+  rw [h₁ n x hn₀]
+  suffices g₀: 1 < f n x + 1 / ↑n
+  . nth_rw 1 [← mul_one (f n x)]
+    refine mul_lt_mul' ?_ g₀ ?_ ?_
+    . exact Preorder.le_refl (f n x)
+    . exact zero_le_one' ℝ
+    . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+  . exact lt_add_of_tsub_lt_right hn₁
+
+
+lemma imo_1985_p6_main_58
+  (f₀ : ℕ → NNReal → NNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n ↦ Function.invFun (f₀ n))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₀ : fb = sn.restrict fun n ↦ fi (↑n) (1 - 1 / ↑↑n))
+  (n : ↑sn):
+   0 ≤ fb n := by
+  have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by
+    rw [hfb₀, hfi]
+    exact rfl
+  rw [hfb₂]
+  simp
+
+
+lemma imo_1985_p6_main_61
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (br : ℝ)
+  (cr : ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hu₅ : br ≤ cr)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x):
+   ∃! a, ∀ (n : ℕ), 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact imo_1985_p6_exists f h₂ hmo₀ f₀ hf₁ sn hsn fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr hfr sbr scr hsbr hscr br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact imo_1985_p6_unique f h₁ hmo₀ h₇ x y hx₀ hy₀
+
+set_option linter.unusedVariables.analyzeTactics true
+
+lemma imo_1985_p6_unique_top_19
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (z : ℝ)
+  (hz₁ : 0 < fd a b i)
+  (j : ℕ)
+  (hj : j = ⌈2 + Real.log (z / fd a b i) / Real.log (3 / 2)⌉₊)
+  (hj₁ : j ∈ sd)
+  (k : ↑sd)
+  (hk₀ : ⟨j, hj₁⟩ ≤ k)
+  (hk₁ : fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k)
+  (hk₂ : i < k):
+   z ≤ fd a b k := by
+  refine le_trans ?_ hk₁
+  refine (div_le_iff₀' ?_).mp ?_
+  . exact hz₁
+  . refine Real.le_pow_of_log_le (by linarith) ?_
+    refine (div_le_iff₀ ?_).mp ?_
+    . refine Real.log_pos ?_
+      linarith
+    . rw [Nat.cast_sub ?_]
+      . rw [Nat.cast_two]
+        refine le_sub_iff_add_le'.mpr ?_
+        refine Nat.le_of_ceil_le ?_
+        exact le_of_eq_of_le (id (Eq.symm hj)) hk₀
+      . rw [hi₁] at hk₂
+        exact Nat.le_of_succ_le hk₂
+
+lemma imo_1985_p6_unique_top_20
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (z : ℝ)
+  (j : ℕ)
+  (hj : j = ⌈2 + Real.log (z / fd a b i) / Real.log (3 / 2)⌉₊)
+  (hj₁ : j ∈ sd)
+  (k : ↑sd)
+  (hk₀ : ⟨j, hj₁⟩ ≤ k)
+  (hk₂ : i < k):
+   z / fd a b i ≤ (3 / 2) ^ (k.1 - 2) := by
+  refine Real.le_pow_of_log_le (by linarith) ?_
+  refine (div_le_iff₀ ?_).mp ?_
+  . refine Real.log_pos ?_
+    linarith
+  . rw [Nat.cast_sub ?_]
+    . rw [Nat.cast_two]
+      refine le_sub_iff_add_le'.mpr ?_
+      refine Nat.le_of_ceil_le ?_
+      exact le_of_eq_of_le (id (Eq.symm hj)) hk₀
+    . rw [hi₁] at hk₂
+      exact Nat.le_of_succ_le hk₂
+
+lemma imo_1985_p6_unique_top_21
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (z : ℝ)
+  (j : ℕ)
+  (hj : j = ⌈2 + Real.log (z / fd a b i) / Real.log (3 / 2)⌉₊)
+  (hj₁ : j ∈ sd)
+  (k : ↑sd)
+  (hk₀ : ⟨j, hj₁⟩ ≤ k)
+  (hk₂ : i < k):
+   Real.log (z / fd a b i) ≤ ↑(k.1 - 2) * Real.log (3 / 2) := by
+  refine (div_le_iff₀ ?_).mp ?_
+  . refine Real.log_pos ?_
+    linarith
+  . rw [Nat.cast_sub ?_]
+    . rw [Nat.cast_two]
+      refine le_sub_iff_add_le'.mpr ?_
+      refine Nat.le_of_ceil_le ?_
+      exact le_of_eq_of_le (id (Eq.symm hj)) hk₀
+    . rw [hi₁] at hk₂
+      exact Nat.le_of_succ_le hk₂
+
+
+lemma imo_1985_p6_unique_top_22
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (z : ℝ)
+  (j : ℕ)
+  (hj : j = ⌈2 + Real.log (z / fd a b i) / Real.log (3 / 2)⌉₊)
+  (hj₁ : j ∈ sd)
+  (k : ↑sd)
+  (hk₀ : ⟨j, hj₁⟩ ≤ k)
+  (hk₂ : i < k):
+   Real.log (z / fd a b i) / Real.log (3 / 2) ≤ ↑(k.1 - 2) := by
+  rw [Nat.cast_sub ?_]
+  . rw [Nat.cast_two]
+    refine le_sub_iff_add_le'.mpr ?_
+    refine Nat.le_of_ceil_le ?_
+    exact le_of_eq_of_le (id (Eq.symm hj)) hk₀
+  . rw [hi₁] at hk₂
+    exact Nat.le_of_succ_le hk₂
+
+
+
+
+lemma imo_1985_p6_unique_nhds
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n) :
+  ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1) →
+        Filter.Tendsto (fd a b) Filter.atTop (nhds 0) := by
+  intros a b ha₀ ha₁
+  have hsd₁: Nonempty ↑sd := by
+    rw [hsd]
+    refine Set.Nonempty.to_subtype ?_
+    exact Set.nonempty_Ici
+  refine tendsto_atTop_nhds.mpr ?_
+  intros U hU₀ hU₁
+  have hU₂: U ∈ nhds 0 := by exact IsOpen.mem_nhds hU₁ hU₀
+  apply mem_nhds_iff_exists_Ioo_subset.mp at hU₂
+  obtain ⟨l, u, hl₀, hl₁⟩ := hU₂
+  have hl₂: 0 < u := by exact (Set.mem_Ioo.mpr hl₀).2
+  let nd := 2 + Nat.ceil (1/u)
+  have hnd₀: nd ∈ sd := by
+    rw [hsd]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right 2 ⌈1 / u⌉₊
+  use ⟨nd, hnd₀⟩
+  intros n hn₀
+  refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+  refine mem_nhds_iff.mpr ?_
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+
+lemma imo_1985_p6_unique_nhds_1
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2):
+   Nonempty ↑sd := by
+  rw [hsd]
+  refine Set.Nonempty.to_subtype ?_
+  exact Set.nonempty_Ici
+
+
+lemma imo_1985_p6_unique_nhds_2
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (hsd₁ : Nonempty ↑sd):
+   Filter.Tendsto (fd a b) Filter.atTop (nhds 0) := by
+  refine tendsto_atTop_nhds.mpr ?_
+  intros U hU₀ hU₁
+  have hU₂: U ∈ nhds 0 := by exact IsOpen.mem_nhds hU₁ hU₀
+  apply mem_nhds_iff_exists_Ioo_subset.mp at hU₂
+  obtain ⟨l, u, hl₀, hl₁⟩ := hU₂
+  have hl₂: 0 < u := by exact (Set.mem_Ioo.mpr hl₀).2
+  let nd := 2 + Nat.ceil (1/u)
+  have hnd₀: nd ∈ sd := by
+    rw [hsd]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right 2 ⌈1 / u⌉₊
+  use ⟨nd, hnd₀⟩
+  intros n hn₀
+  refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+  refine mem_nhds_iff.mpr ?_
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+
+lemma imo_1985_p6_unique_nhds_3
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (U : Set ℝ)
+  (hU₀ : 0 ∈ U)
+  (hU₁ : IsOpen U):
+   ∃ N, ∀ (n : ↑sd), N ≤ n → fd a b n ∈ U := by
+  have hU₂: U ∈ nhds 0 := by exact IsOpen.mem_nhds hU₁ hU₀
+  apply mem_nhds_iff_exists_Ioo_subset.mp at hU₂
+  obtain ⟨l, u, hl₀, hl₁⟩ := hU₂
+  have hl₂: 0 < u := by exact (Set.mem_Ioo.mpr hl₀).2
+  let nd := 2 + Nat.ceil (1/u)
+  have hnd₀: nd ∈ sd := by
+    rw [hsd]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right 2 ⌈1 / u⌉₊
+  use ⟨nd, hnd₀⟩
+  intros n hn₀
+  refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+  refine mem_nhds_iff.mpr ?_
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+
+lemma imo_1985_p6_unique_nhds_4
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (U : Set ℝ)
+  (hU₁ : IsOpen U)
+  (hU₂ : ∃ l u, 0 ∈ Set.Ioo l u ∧ Set.Ioo l u ⊆ U):
+   ∃ N, ∀ (n : ↑sd), N ≤ n → fd a b n ∈ U := by
+  obtain ⟨l, u, hl₀, hl₁⟩ := hU₂
+  have hl₂: 0 < u := by exact (Set.mem_Ioo.mpr hl₀).2
+  let nd := 2 + Nat.ceil (1/u)
+  have hnd₀: nd ∈ sd := by
+    rw [hsd]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right 2 ⌈1 / u⌉₊
+  use ⟨nd, hnd₀⟩
+  intros n hn₀
+  refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+  refine mem_nhds_iff.mpr ?_
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+
+lemma imo_1985_p6_unique_nhds_5
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (U : Set ℝ)
+  (hU₁ : IsOpen U)
+  (l u : ℝ)
+  (hl₀ : 0 ∈ Set.Ioo l u)
+  (hl₁ : Set.Ioo l u ⊆ U):
+   ∃ N, ∀ (n : ↑sd), N ≤ n → fd a b n ∈ U := by
+  have hl₂: 0 < u := by exact (Set.mem_Ioo.mpr hl₀).2
+  let nd := 2 + Nat.ceil (1/u)
+  have hnd₀: nd ∈ sd := by
+    rw [hsd]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right 2 ⌈1 / u⌉₊
+  use ⟨nd, hnd₀⟩
+  intros n hn₀
+  refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+  refine mem_nhds_iff.mpr ?_
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+
+lemma imo_1985_p6_unique_nhds_6
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (u : ℝ)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊):
+  nd ∈ sd := by
+  rw [hsd, hnd]
+  refine Set.mem_Ici.mpr ?_
+  exact Nat.le_add_right 2 ⌈1 / u⌉₊
+
+lemma imo_1985_p6_unique_nhds_7
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (U : Set ℝ)
+  (hU₁ : IsOpen U)
+  (l u : ℝ)
+  (hl₀ : 0 ∈ Set.Ioo l u)
+  (hl₁ : Set.Ioo l u ⊆ U)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (hnd₀ : nd ∈ sd):
+   ∃ N, ∀ (n : ↑sd), N ≤ n → fd a b n ∈ U := by
+  use ⟨nd, hnd₀⟩
+  intros n hn₀
+  refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+  refine mem_nhds_iff.mpr ?_
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        rw [hnd]
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+
+lemma imo_1985_p6_unique_nhds_8
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (U : Set ℝ)
+  (hU₁ : IsOpen U)
+  (l u : ℝ)
+  (hl₀ : 0 ∈ Set.Ioo l u)
+  (hl₁ : Set.Ioo l u ⊆ U)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (hnd₀ : nd ∈ sd)
+  (n : ↑sd)
+  (hn₀ : ⟨nd, hnd₀⟩ ≤ n):
+   fd a b n ∈ U := by
+  refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+  refine mem_nhds_iff.mpr ?_
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        rw [hnd]
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+lemma imo_1985_p6_unique_nhds_9
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (U : Set ℝ)
+  (l u : ℝ)
+  (hl₀ : 0 ∈ Set.Ioo l u)
+  (hl₁ : Set.Ioo l u ⊆ U)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (hnd₀ : nd ∈ sd)
+  (n : ↑sd)
+  (hn₀ : ⟨nd, hnd₀⟩ ≤ n):
+   U ∈ nhds (fd a b n) := by
+  refine mem_nhds_iff.mpr ?_
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        rw [hnd]
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+lemma imo_1985_p6_unique_nhds_10
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (U : Set ℝ)
+  (l u : ℝ)
+  (hl₀ : 0 ∈ Set.Ioo l u)
+  (hl₁ : Set.Ioo l u ⊆ U)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (hnd₀ : nd ∈ sd)
+  (n : ↑sd)
+  (hn₀ : ⟨nd, hnd₀⟩ ≤ n):
+   ∃ t ⊆ U, IsOpen t ∧ fd a b n ∈ t := by
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        rw [hnd]
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+lemma imo_1985_p6_unique_nhds_11
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (U : Set ℝ)
+  (hU₁ : IsOpen U)
+  (nd : ℕ)
+  (hnd₀ : nd ∈ sd)
+  (hnd₁: ∀ n:↑ sd, ∃ t ⊆ U, IsOpen t ∧ fd a b n ∈ t):
+   ∃ N, ∀ (n : ↑sd), N ≤ n → fd a b n ∈ U := by
+  use ⟨nd, hnd₀⟩
+  intros n _
+  refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+  refine mem_nhds_iff.mpr ?_
+  exact hnd₁ n
+
+lemma imo_1985_p6_unique_nhds_12
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (U : Set ℝ)
+  (l u : ℝ)
+  (hl₀ : 0 ∈ Set.Ioo l u)
+  (hl₁ : Set.Ioo l u ⊆ U)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (hnd₀ : nd ∈ sd)
+  (n : ↑sd)
+  (hn₀ : ⟨nd, hnd₀⟩ ≤ n):
+   Set.Ioo l u ⊆ U ∧ IsOpen (Set.Ioo l u) ∧ fd a b n ∈ Set.Ioo l u := by
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        rw [hnd]
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+
+lemma imo_1985_p6_unique_nhds_13
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (l u : ℝ)
+  (hl₀ : 0 ∈ Set.Ioo l u)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (hnd₀ : nd ∈ sd)
+  (n : ↑sd)
+  (hn₀ : ⟨nd, hnd₀⟩ ≤ n):
+   fd a b n ∈ Set.Ioo l u := by
+  refine Set.mem_Ioo.mpr ?_
+  constructor
+  . refine lt_trans ?_ (hd₁ n a b ha₀)
+    exact (Set.mem_Ioo.mp hl₀).1
+  . have hn₁: fd a b n < 1 / n := by
+      rw [hfd₁]
+      have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+      have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+      refine sub_lt_iff_lt_add.mpr ?_
+      refine lt_trans hb₁ ?_
+      exact sub_lt_iff_lt_add'.mp ha₂
+    have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+      refine one_div_le_one_div_of_le ?_ ?_
+      . refine Nat.cast_pos.mpr ?_
+        rw [hsd] at hnd₀
+        exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+      . exact Nat.cast_le.mpr hn₀
+    refine lt_of_lt_of_le hn₁ ?_
+    refine le_trans hn₂ ?_
+    refine div_le_of_le_mul₀ ?_ ?_ ?_
+    . exact Nat.cast_nonneg' nd
+    . exact le_of_lt hl₂
+    . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+        refine (mul_le_mul_left hl₂).mpr ?_
+        rw [Nat.cast_add 2 _, Nat.cast_two]
+        refine add_le_add_left ?_ 2
+        exact Nat.le_ceil (1 / u)
+      rw [hnd]
+      refine le_trans ?_ hl₃
+      rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+      refine le_of_lt ?_
+      refine sub_lt_iff_lt_add.mp ?_
+      rw [sub_self 1]
+      exact mul_pos hl₂ (two_pos)
+
+
+lemma imo_1985_p6_unique_nhds_14
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (l u : ℝ)
+  (hl₀ : 0 ∈ Set.Ioo l u)
+  (n : ↑sd):
+   l < fd a b n := by
+  refine lt_trans ?_ (hd₁ n a b ha₀)
+  exact (Set.mem_Ioo.mp hl₀).1
+
+lemma imo_1985_p6_unique_nhds_15
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (a b : NNReal)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (u : ℝ)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (hnd₀ : nd ∈ sd)
+  (n : ↑sd)
+  (hn₀ : ⟨nd, hnd₀⟩ ≤ n):
+   fd a b n < u := by
+  have hn₁: fd a b n < 1 / n := by
+    rw [hfd₁]
+    have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+    have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+    refine sub_lt_iff_lt_add.mpr ?_
+    refine lt_trans hb₁ ?_
+    exact sub_lt_iff_lt_add'.mp ha₂
+  have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+    refine one_div_le_one_div_of_le ?_ ?_
+    . refine Nat.cast_pos.mpr ?_
+      rw [hsd] at hnd₀
+      exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+    . exact Nat.cast_le.mpr hn₀
+  refine lt_of_lt_of_le hn₁ ?_
+  refine le_trans hn₂ ?_
+  refine div_le_of_le_mul₀ ?_ ?_ ?_
+  . exact Nat.cast_nonneg' nd
+  . exact le_of_lt hl₂
+  . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+      refine (mul_le_mul_left hl₂).mpr ?_
+      rw [Nat.cast_add 2 _, Nat.cast_two]
+      refine add_le_add_left ?_ 2
+      exact Nat.le_ceil (1 / u)
+    rw [hnd]
+    refine le_trans ?_ hl₃
+    rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+    refine le_of_lt ?_
+    refine sub_lt_iff_lt_add.mp ?_
+    rw [sub_self 1]
+    exact mul_pos hl₂ (two_pos)
+
+
+
+lemma imo_1985_p6_unique_nhds_16
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (a b : NNReal)
+  (ha₁ : ∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1)
+  (n : ↑sd):
+   fd a b n < 1 / ↑↑n := by
+  rw [hfd₁]
+  have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+  have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+  refine sub_lt_iff_lt_add.mpr ?_
+  refine lt_trans hb₁ ?_
+  exact sub_lt_iff_lt_add'.mp ha₂
+
+
+lemma imo_1985_p6_unique_nhds_17
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (u : ℝ)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (hnd₀ : nd ∈ sd)
+  (n : ↑sd)
+  (hn₀ : ⟨nd, hnd₀⟩ ≤ n)
+  (hn₁ : fd a b n < 1 / ↑↑n):
+   fd a b n < u := by
+  have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+    refine one_div_le_one_div_of_le ?_ ?_
+    . refine Nat.cast_pos.mpr ?_
+      rw [hsd] at hnd₀
+      exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+    . exact Nat.cast_le.mpr hn₀
+  refine lt_of_lt_of_le hn₁ ?_
+  refine le_trans hn₂ ?_
+  refine div_le_of_le_mul₀ ?_ ?_ ?_
+  . exact Nat.cast_nonneg' nd
+  . exact le_of_lt hl₂
+  . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+      refine (mul_le_mul_left hl₂).mpr ?_
+      rw [Nat.cast_add 2 _, Nat.cast_two]
+      refine add_le_add_left ?_ 2
+      exact Nat.le_ceil (1 / u)
+    rw [hnd]
+    refine le_trans ?_ hl₃
+    rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+    refine le_of_lt ?_
+    refine sub_lt_iff_lt_add.mp ?_
+    rw [sub_self 1]
+    exact mul_pos hl₂ (two_pos)
+
+
+
+lemma imo_1985_p6_unique_nhds_18
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (nd : ℕ)
+  (hnd₀ : nd ∈ sd)
+  (n : ↑sd)
+  (hn₀ : ⟨nd, hnd₀⟩ ≤ n):
+   (1:ℝ) / n ≤ 1 / nd := by
+  refine one_div_le_one_div_of_le ?_ ?_
+  . refine Nat.cast_pos.mpr ?_
+    rw [hsd] at hnd₀
+    exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+  . exact Nat.cast_le.mpr hn₀
+
+lemma imo_1985_p6_unique_nhds_19
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (u : ℝ)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (n : ↑sd)
+  (hn₁ : fd a b n < 1 / ↑↑n)
+  (hn₂ : (1:ℝ) / ↑↑n ≤ 1 / ↑nd):
+   fd a b n < u := by
+  refine lt_of_lt_of_le hn₁ ?_
+  refine le_trans hn₂ ?_
+  refine div_le_of_le_mul₀ ?_ ?_ ?_
+  . exact Nat.cast_nonneg' nd
+  . exact le_of_lt hl₂
+  . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+      refine (mul_le_mul_left hl₂).mpr ?_
+      rw [Nat.cast_add 2 _, Nat.cast_two]
+      refine add_le_add_left ?_ 2
+      exact Nat.le_ceil (1 / u)
+    rw [hnd]
+    refine le_trans ?_ hl₃
+    rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+    refine le_of_lt ?_
+    refine sub_lt_iff_lt_add.mp ?_
+    rw [sub_self 1]
+    exact mul_pos hl₂ (two_pos)
+
+lemma imo_1985_p6_unique_nhds_20
+  (u : ℝ)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊):
+   1 / ↑nd ≤ u := by
+  refine div_le_of_le_mul₀ ?_ ?_ ?_
+  . exact Nat.cast_nonneg' nd
+  . exact le_of_lt hl₂
+  . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+      refine (mul_le_mul_left hl₂).mpr ?_
+      rw [Nat.cast_add 2 _, Nat.cast_two]
+      refine add_le_add_left ?_ 2
+      exact Nat.le_ceil (1 / u)
+    rw [hnd]
+    refine le_trans ?_ hl₃
+    rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+    refine le_of_lt ?_
+    refine sub_lt_iff_lt_add.mp ?_
+    rw [sub_self 1]
+    exact mul_pos hl₂ (two_pos)
+
+lemma imo_1985_p6_unique_nhds_21
+  (u : ℝ)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊):
+   1 ≤ u * ↑nd := by
+  have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+    refine (mul_le_mul_left hl₂).mpr ?_
+    rw [Nat.cast_add 2 _, Nat.cast_two]
+    refine add_le_add_left ?_ 2
+    exact Nat.le_ceil (1 / u)
+  rw [hnd]
+  refine le_trans ?_ hl₃
+  rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+  refine le_of_lt ?_
+  refine sub_lt_iff_lt_add.mp ?_
+  rw [sub_self 1]
+  exact mul_pos hl₂ (two_pos)
+
+
+lemma imo_1985_p6_unique_nhds_22
+  (u : ℝ)
+  (hl₂ : 0 < u):
+   u * (2 + 1 / u) ≤ u * ↑(2 + ⌈1 / u⌉₊) := by
+  refine (mul_le_mul_left hl₂).mpr ?_
+  rw [Nat.cast_add 2 _, Nat.cast_two]
+  refine add_le_add_left ?_ 2
+  exact Nat.le_ceil (1 / u)
+
+lemma imo_1985_p6_unique_nhds_23
+  (u : ℝ)
+  (hl₂ : 0 < u)
+  (nd : ℕ)
+  (hnd : nd = 2 + ⌈1 / u⌉₊)
+  (hl₃ : u * (2 + 1 / u) ≤ u * ↑(2 + ⌈1 / u⌉₊)):
+   1 ≤ u * ↑nd := by
+  rw [hnd]
+  refine le_trans ?_ hl₃
+  rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+  refine le_of_lt ?_
+  refine sub_lt_iff_lt_add.mp ?_
+  rw [sub_self 1]
+  exact mul_pos hl₂ (two_pos)
+
+lemma imo_1985_p6_unique_top_ind
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hd₃: ∀ (nd : ↑sd), nd.1 + (1:ℕ) ∈ sd)
+  (hd₂ : ∀ (nd : ↑sd), fd a b nd * (2 - 1 / ↑↑nd) ≤ fd a b ⟨nd.1 + 1, hd₃ nd⟩)
+  (hi₀ : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi₀⟩) :
+  ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd := by
+  intro nd
+  rw [hfd₁ a b nd]
+  have hnd₀: 2 ≤ nd.1 := by
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  refine Nat.le_induction ?_ ?_ nd.1 hnd₀
+  . have hi₂: i.val = (2:ℕ) := by
+      simp_all only [Subtype.forall]
+    rw [hfd₁ a b i, hi₂]
+    simp
+  . simp
+    intros n hn₀ hn₁
+    have hn₂: n - 1 = n - 2 + 1 := by
+      simp
+      exact (Nat.sub_eq_iff_eq_add hn₀).mp rfl
+    have hn₃: n ∈ sd := by
+      rw [hsd]
+      exact hn₀
+    let nn : ↑sd := ⟨n, hn₃⟩
+    have hnn: nn.1 = n := by exact rfl
+    have hn₄: nn.1 + 1 ∈ sd := by
+      rw [hnn, hsd]
+      refine Set.mem_Ici.mpr ?_
+      exact Nat.le_add_right_of_le hn₀
+    have hn₅: fd a b nn * (2 - 1 / ↑n) ≤ fd a b ⟨nn.1 + 1, hn₄⟩ := by exact hd₂ nn
+    rw [hfd₁ a b ⟨nn.1 + 1, hn₄⟩] at hn₅
+    have hn₆: f (↑nn + 1) b - f (↑nn + 1) a = f (n + 1) b - f (n + 1) a := by exact rfl
+    rw [hn₆] at hn₅
+    refine le_trans ?_ hn₅
+    rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+    refine mul_le_mul ?_ ?_ (by linarith) ?_
+    . refine le_of_le_of_eq hn₁ ?_
+      rw [hfd₁]
+    . refine (div_le_iff₀ (two_pos)).mpr ?_
+      rw [sub_mul, one_div_mul_eq_div _ 2]
+      refine le_sub_iff_add_le.mpr ?_
+      refine le_sub_iff_add_le'.mp ?_
+      refine (div_le_iff₀ ?_).mpr ?_
+      . refine Nat.cast_pos.mpr ?_
+        exact lt_of_lt_of_le (two_pos) hn₀
+      . ring_nf
+        exact Nat.ofNat_le_cast.mpr hn₀
+    . exact le_of_lt (hd₁ nn a b ha₀)
+
+
+
+lemma imo_1985_p6_unique_top
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n) :
+  ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b)
+      → Filter.Tendsto (fd a b) Filter.atTop Filter.atTop := by
+  intros a b ha₀ ha₁
+  have hd₀: ∀ (nd:↑sd), (nd.1 + 1) ∈ sd := by
+    intro nd
+    let t : ℕ := nd.1
+    have ht: t = nd.1 := by rfl
+    rw [← ht, hsd]
+    refine Set.mem_Ici.mpr ?_
+    refine Nat.le_add_right_of_le ?_
+    refine Set.mem_Ici.mp ?_
+    rw [ht, ← hsd]
+    exact nd.2
+  have hd₂: ∀ nd, fd a b nd  * (2 - 1 / nd.1) ≤ fd a b ⟨nd.1 + 1, hd₀ nd⟩ := by
+    intro nd
+    have hnd₀: 0 < nd.1 := by
+      have g₀: 2 ≤ nd.1 := by
+        refine Set.mem_Ici.mp ?_
+        rw [← hsd]
+        exact nd.2
+      exact Nat.zero_lt_of_lt g₀
+    rw [hfd₁, hfd₁, h₁ nd.1 _ hnd₀, h₁ nd.1 _ hnd₀]
+    have hnd₁: f (↑nd) b * (f (↑nd) b + 1 / ↑↑nd) - f (↑nd) a * (f (↑nd) a + 1 / ↑↑nd) =
+      (f (↑nd) b - f (↑nd) a) * (f (↑nd) b + f (↑nd) a + 1 / nd.1) := by
+      ring_nf
+    rw [hnd₁]
+    refine (mul_le_mul_left ?_).mpr ?_
+    . rw [← hfd₁]
+      exact hd₁ nd a b ha₀
+    . refine le_sub_iff_add_le.mp ?_
+      rw [sub_neg_eq_add]
+      have hnd₂: 1 - 1 / nd.1 < f (↑nd) b := by
+        exact h₇ nd.1 b hnd₀ (ha₁ nd).2
+      have hnd₃: 1 - 1 / nd.1 < f (↑nd) a := by
+        exact h₇ nd.1 a hnd₀ (ha₁ nd).1
+      linarith
+  have hi: 2 ∈ sd := by
+    rw [hsd]
+    decide
+  let i : ↑sd := ⟨(2:ℕ), hi⟩
+  have hd₃: ∀ nd, fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd := by
+    intro nd
+    exact imo_1985_p6_unique_top_ind f sd hsd fd hfd₁ hd₁ a b ha₀ hd₀ hd₂ hi i rfl nd
+  have hsd₁: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    exact Set.nonempty_of_mem (hd₀ i)
+  refine Filter.tendsto_atTop_atTop.mpr ?_
+  intro z
+  by_cases hz₀: z ≤ fd a b i
+  . use i
+    intros j _
+    refine le_trans hz₀ ?_
+    refine le_trans ?_ (hd₃ j)
+    refine le_mul_of_one_le_right ?_ ?_
+    . refine le_of_lt ?_
+      exact hd₁ i a b ha₀
+    . refine one_le_pow₀ ?_
+      linarith
+  . push_neg at hz₀
+    have hz₁: 0 < fd a b i := by exact hd₁ i a b ha₀
+    have hz₂: 0 < Real.log (z / fd a b i) := by
+      refine Real.log_pos ?_
+      exact (one_lt_div hz₁).mpr hz₀
+    let j : ℕ := Nat.ceil (2 + Real.log (z / fd a b i) / Real.log (3 / 2))
+    have hj₀: 2 < j := by
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    have hj₁: j ∈ sd := by
+      rw [hsd]
+      exact Set.mem_Ici_of_Ioi hj₀
+    use ⟨j, hj₁⟩
+    intro k hk₀
+    have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+      exact hd₃ k
+    have hk₂: i < k := by
+      refine lt_of_lt_of_le ?_ hk₀
+      refine Subtype.mk_lt_mk.mpr ?_
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    refine le_trans ?_ hk₁
+    refine (div_le_iff₀' ?_).mp ?_
+    . exact hz₁
+    . refine Real.le_pow_of_log_le (by linarith) ?_
+      refine (div_le_iff₀ ?_).mp ?_
+      . refine Real.log_pos ?_
+        linarith
+      . rw [Nat.cast_sub ?_]
+        . rw [Nat.cast_two]
+          refine le_sub_iff_add_le'.mpr ?_
+          exact Nat.le_of_ceil_le hk₀
+        . exact Nat.le_of_succ_le hk₂
+
+
+lemma imo_1985_p6_unique_1
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁):
+   ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n := by
+  intros nd a b hnd₀
+  rw [hfd₁]
+  refine sub_pos.mpr ?_
+  refine hmo₀ nd.1 ?_ hnd₀
+  refine lt_of_lt_of_le (Nat.zero_lt_two) ?_
+  refine Set.mem_Ici.mp ?_
+  rw [← hsd]
+  exact Subtype.coe_prop nd
+
+
+
+lemma imo_1985_p6_unique_2
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (x y : NNReal)
+  (hx₀ : ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1)
+  (hy₀ : ∀ (n : ℕ), 0 < n → 0 < f n y ∧ f n y < f (n + 1) y ∧ f (n + 1) y < 1)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n):
+   x = y := by
+  have hfd₂: ∀ a b, a < b → (∀ n:↑sd, f n.1 a < f (n.1 + 1) a ∧ f n.1 b < f (n.1 + 1) b)
+      → Filter.Tendsto (fd a b) Filter.atTop Filter.atTop := by
+    intros a b ha₀ ha₁
+    exact imo_1985_p6_unique_top f h₁ h₇ sd hsd fd hfd₁ hd₁ a b ha₀ ha₁
+  have hfd₃: ∀ a b, a < b → (∀ (n:↑sd), (1 - 1 / n.1 < f n.1 a ∧ 1 - 1 / n.1 < f n.1 b) ∧ (f n.1 a < 1 ∧ f n.1 b < 1))
+      → Filter.Tendsto (fd a b) Filter.atTop (nhds 0) := by
+    intros a b ha₀ ha₁
+    exact imo_1985_p6_unique_nhds f sd hsd fd hfd₁ hd₁ a b ha₀ ha₁
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hd₂: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    rw [hsd]
+    exact Set.nonempty_Ici
+  by_contra! hc₀
+  by_cases hy₁: x < y
+  . have hy₂: Filter.Tendsto (fd x y) Filter.atTop Filter.atTop := by
+      refine hfd₂ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) (hd₁ nd)
+      constructor
+      . exact (hx₀ nd.1 hnd₀).2.1
+      . exact (hy₀ nd.1 hnd₀).2.1
+    have hy₃: Filter.Tendsto (fd x y) Filter.atTop (nhds 0) := by
+      refine hfd₃ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by
+        refine lt_of_lt_of_le ?_ (hd₁ nd)
+        exact Nat.zero_lt_two
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        refine lt_of_lt_of_le ?_ (hd₁ nd)
+        exact Nat.one_lt_two
+      constructor
+      . constructor
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₂
+    apply tendsto_atTop_nhds.mp at hy₃
+    contrapose! hy₃
+    clear hy₃
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        nth_rw 1 [hsd]
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact (hd₁ N)
+        . refine le_trans ?_ (hd₁ i)
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd x y a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+  . have hy₂: y < x := by
+      push_neg at hy₁
+      exact lt_of_le_of_ne hy₁ hc₀.symm
+    have hy₃: Filter.Tendsto (fd y x) Filter.atTop Filter.atTop := by
+      refine hfd₂ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) (hd₁ nd)
+      constructor
+      . exact (hy₀ nd.1 hnd₀).2.1
+      . exact (hx₀ nd.1 hnd₀).2.1
+    have hy₄: Filter.Tendsto (fd y x) Filter.atTop (nhds 0) := by
+      refine hfd₃ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (Nat.zero_lt_two) (hd₁ nd)
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        exact lt_of_lt_of_le (Nat.one_lt_two) (hd₁ nd)
+      constructor
+      . constructor
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₃
+    apply tendsto_atTop_nhds.mp at hy₄
+    contrapose! hy₄
+    clear hy₄
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd y x a := by exact hy₃ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        nth_rw 1 [hsd]
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact hd₁ N
+        . refine le_trans ?_ (hd₁ i)
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd y x a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+
+
+
+
+lemma imo_1985_p6_unique_3
+  (f : ℕ → NNReal → ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (x y : NNReal)
+  (hx₀ : ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1)
+  (hy₀ : ∀ (n : ℕ), 0 < n → 0 < f n y ∧ f n y < f (n + 1) y ∧ f (n + 1) y < 1)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₂ : ∀ (a b : NNReal),
+    (a < b →
+      (∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b) → Filter.Tendsto (fd a b) Filter.atTop Filter.atTop))
+  (hfd₃ : ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1) →
+        (Filter.Tendsto (fd a b) Filter.atTop (nhds 0)) ):
+   x = y := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hd₂: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    rw [hsd]
+    exact Set.nonempty_Ici
+  by_contra! hc₀
+  by_cases hy₁: x < y
+  . have hy₂: Filter.Tendsto (fd x y) Filter.atTop Filter.atTop := by
+      refine hfd₂ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) (hd₁ nd)
+      constructor
+      . exact (hx₀ nd.1 hnd₀).2.1
+      . exact (hy₀ nd.1 hnd₀).2.1
+    have hy₃: Filter.Tendsto (fd x y) Filter.atTop (nhds 0) := by
+      refine hfd₃ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by
+        refine lt_of_lt_of_le ?_ (hd₁ nd)
+        exact Nat.zero_lt_two
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        refine lt_of_lt_of_le ?_ (hd₁ nd)
+        exact Nat.one_lt_two
+      constructor
+      . constructor
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₂
+    apply tendsto_atTop_nhds.mp at hy₃
+    contrapose! hy₃
+    clear hy₃
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        nth_rw 1 [hsd]
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact hd₁ N
+        . refine le_trans ?_ (hd₁ i)
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd x y a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+  . have hy₂: y < x := by
+      push_neg at hy₁
+      exact lt_of_le_of_ne hy₁ hc₀.symm
+    have hy₃: Filter.Tendsto (fd y x) Filter.atTop Filter.atTop := by
+      refine hfd₂ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) (hd₁ nd)
+      constructor
+      . exact (hy₀ nd.1 hnd₀).2.1
+      . exact (hx₀ nd.1 hnd₀).2.1
+    have hy₄: Filter.Tendsto (fd y x) Filter.atTop (nhds 0) := by
+      refine hfd₃ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (Nat.zero_lt_two) (hd₁ nd)
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        exact lt_of_lt_of_le (Nat.one_lt_two) (hd₁ nd)
+      constructor
+      . constructor
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₃
+    apply tendsto_atTop_nhds.mp at hy₄
+    contrapose! hy₄
+    clear hy₄
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd y x a := by exact hy₃ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        nth_rw 1 [hsd]
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact hd₁ N
+        . refine le_trans ?_ (hd₁ i)
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd y x a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+
+
+
+lemma imo_1985_p6_unique_4
+  (f : ℕ → NNReal → ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (x y : NNReal)
+  (hx₀ : ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1)
+  (hy₀ : ∀ (n : ℕ), 0 < n → 0 < f n y ∧ f n y < f (n + 1) y ∧ f (n + 1) y < 1)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₂ : ∀ (a b : NNReal),
+    a < b →
+      ((∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b) → Filter.Tendsto (fd a b) Filter.atTop Filter.atTop))
+  (hfd₃ : ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1) →
+        (Filter.Tendsto (fd a b) Filter.atTop (nhds 0)))
+  (hc₀ : x ≠ y):
+   False := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hd₂: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    rw [hsd]
+    exact Set.nonempty_Ici
+  by_cases hy₁: x < y
+  . have hy₂: Filter.Tendsto (fd x y) Filter.atTop Filter.atTop := by
+      refine hfd₂ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) (hd₁ nd)
+      constructor
+      . exact (hx₀ nd.1 hnd₀).2.1
+      . exact (hy₀ nd.1 hnd₀).2.1
+    have hy₃: Filter.Tendsto (fd x y) Filter.atTop (nhds 0) := by
+      refine hfd₃ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by
+        refine lt_of_lt_of_le ?_ (hd₁ nd)
+        exact Nat.zero_lt_two
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        refine lt_of_lt_of_le ?_ (hd₁ nd)
+        exact Nat.one_lt_two
+      constructor
+      . constructor
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₂
+    apply tendsto_atTop_nhds.mp at hy₃
+    contrapose! hy₃
+    clear hy₃
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        nth_rw 1 [hsd]
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact (hd₁ N)
+        . refine le_trans ?_ (hd₁ i)
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd x y a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+  . have hy₂: y < x := by
+      push_neg at hy₁
+      exact lt_of_le_of_ne hy₁ hc₀.symm
+    have hy₃: Filter.Tendsto (fd y x) Filter.atTop Filter.atTop := by
+      refine hfd₂ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) (hd₁ nd)
+      constructor
+      . exact (hy₀ nd.1 hnd₀).2.1
+      . exact (hx₀ nd.1 hnd₀).2.1
+    have hy₄: Filter.Tendsto (fd y x) Filter.atTop (nhds 0) := by
+      refine hfd₃ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (Nat.zero_lt_two) (hd₁ nd)
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        exact lt_of_lt_of_le (Nat.one_lt_two) (hd₁ nd)
+      constructor
+      . constructor
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₃
+    apply tendsto_atTop_nhds.mp at hy₄
+    contrapose! hy₄
+    clear hy₄
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd y x a := by exact hy₃ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        nth_rw 1 [hsd]
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact hd₁ N
+        . refine le_trans ?_ (hd₁ i)
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd y x a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+
+
+
+lemma imo_1985_p6_unique_5
+  (f : ℕ → NNReal → ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (x y : NNReal)
+  (hx₀ : ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1)
+  (hy₀ : ∀ (n : ℕ), 0 < n → 0 < f n y ∧ f n y < f (n + 1) y ∧ f (n + 1) y < 1)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₂ : ∀ (a b : NNReal),
+    a < b →
+      ((∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b) → Filter.Tendsto (fd a b) Filter.atTop Filter.atTop))
+  (hfd₃ : ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1) →
+        (Filter.Tendsto (fd a b) Filter.atTop (nhds 0)))
+  (hy₁ : x < y):
+   False := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hd₂: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    rw [hsd]
+    exact Set.nonempty_Ici
+  have hy₂: Filter.Tendsto (fd x y) Filter.atTop Filter.atTop := by
+    refine hfd₂ x y hy₁ ?_
+    intro nd
+    have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) (hd₁ nd)
+    constructor
+    . exact (hx₀ nd.1 hnd₀).2.1
+    . exact (hy₀ nd.1 hnd₀).2.1
+  have hy₃: Filter.Tendsto (fd x y) Filter.atTop (nhds 0) := by
+    refine hfd₃ x y hy₁ ?_
+    intro nd
+    have hnd₀: 0 < nd.1 := by
+      refine lt_of_lt_of_le ?_ (hd₁ nd)
+      exact Nat.zero_lt_two
+    have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+    have hnd₂: 0 < nd.1 - 1 := by
+      refine Nat.sub_pos_of_lt ?_
+      refine lt_of_lt_of_le ?_ (hd₁ nd)
+      exact Nat.one_lt_two
+    constructor
+    . constructor
+      . refine h₇ nd.1 x hnd₀ ?_
+        exact (hx₀ (nd.1) hnd₀).2.1
+      . refine h₇ nd.1 y hnd₀ ?_
+        exact (hy₀ (nd.1) hnd₀).2.1
+    . constructor
+      . rw [← hnd₁]
+        exact (hx₀ (nd.1 - 1) hnd₂).2.2
+      . rw [← hnd₁]
+        exact (hy₀ (nd.1 - 1) hnd₂).2.2
+  apply Filter.tendsto_atTop_atTop.mp at hy₂
+  apply tendsto_atTop_nhds.mp at hy₃
+  contrapose! hy₃
+  clear hy₃
+  let sx : Set ℝ := Set.Ioo (-1) 1
+  use sx
+  constructor
+  . refine Set.mem_Ioo.mpr ?_
+    simp
+  constructor
+  . exact isOpen_Ioo
+  . intro N
+    have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+    obtain ⟨i, hi₀⟩ := hy₅
+    have hi₁: (N.1 + i.1) ∈ sd := by
+      nth_rw 1 [hsd]
+      refine Set.mem_Ici.mpr ?_
+      rw [← add_zero 2]
+      refine Nat.add_le_add ?_ ?_
+      . exact hd₁ N
+      . refine le_trans ?_ (hd₁ i)
+        exact Nat.zero_le 2
+    let a : ↑sd := ⟨N + i, hi₁⟩
+    use a
+    constructor
+    . refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_right ↑N ↑i
+    . refine Set.not_mem_Ioo_of_ge ?_
+      have hi₂: ↑↑N + 3 ≤ fd x y a := by
+        refine hi₀ a ?_
+        refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_left ↑i ↑N
+      refine le_trans ?_ hi₂
+      norm_cast
+      exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_unique_6
+  (f : ℕ → NNReal → ℝ)
+  (x y : NNReal)
+  (hx₀ : ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1)
+  (hy₀ : ∀ (n : ℕ), 0 < n → 0 < f n y ∧ f n y < f (n + 1) y ∧ f (n + 1) y < 1)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₂ : ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b) → Filter.Tendsto (fd a b) Filter.atTop Filter.atTop)
+  (hy₁ : x < y):
+   Filter.Tendsto (fd x y) Filter.atTop Filter.atTop := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  refine hfd₂ x y hy₁ ?_
+  intro nd
+  have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) (hd₁ nd)
+  constructor
+  . exact (hx₀ nd.1 hnd₀).2.1
+  . exact (hy₀ nd.1 hnd₀).2.1
+
+lemma imo_1985_p6_unique_7
+  (f : ℕ → NNReal → ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (x y : NNReal)
+  (hx₀ : ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1)
+  (hy₀ : ∀ (n : ℕ), 0 < n → 0 < f n y ∧ f n y < f (n + 1) y ∧ f (n + 1) y < 1)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₃ : ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1) →
+        (Filter.Tendsto (fd a b) Filter.atTop (nhds 0)))
+  (hy₁ : x < y)
+  (hy₂ : Filter.Tendsto (fd x y) Filter.atTop Filter.atTop):
+   False := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hd₂: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    rw [hsd]
+    exact Set.nonempty_Ici
+  have hy₃: Filter.Tendsto (fd x y) Filter.atTop (nhds 0) := by
+    refine hfd₃ x y hy₁ ?_
+    intro nd
+    have hnd₀: 0 < nd.1 := by
+      refine lt_of_lt_of_le ?_ (hd₁ nd)
+      exact Nat.zero_lt_two
+    have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+    have hnd₂: 0 < nd.1 - 1 := by
+      refine Nat.sub_pos_of_lt ?_
+      refine lt_of_lt_of_le ?_ (hd₁ nd)
+      exact Nat.one_lt_two
+    constructor
+    . constructor
+      . refine h₇ nd.1 x hnd₀ ?_
+        exact (hx₀ (nd.1) hnd₀).2.1
+      . refine h₇ nd.1 y hnd₀ ?_
+        exact (hy₀ (nd.1) hnd₀).2.1
+    . constructor
+      . rw [← hnd₁]
+        exact (hx₀ (nd.1 - 1) hnd₂).2.2
+      . rw [← hnd₁]
+        exact (hy₀ (nd.1 - 1) hnd₂).2.2
+  apply Filter.tendsto_atTop_atTop.mp at hy₂
+  apply tendsto_atTop_nhds.mp at hy₃
+  contrapose! hy₃
+  clear hy₃
+  let sx : Set ℝ := Set.Ioo (-1) 1
+  use sx
+  constructor
+  . refine Set.mem_Ioo.mpr ?_
+    simp
+  constructor
+  . exact isOpen_Ioo
+  . intro N
+    have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+    obtain ⟨i, hi₀⟩ := hy₅
+    have hi₁: (N.1 + i.1) ∈ sd := by
+      nth_rw 1 [hsd]
+      refine Set.mem_Ici.mpr ?_
+      rw [← add_zero 2]
+      refine Nat.add_le_add ?_ ?_
+      . exact hd₁ N
+      . refine le_trans ?_ (hd₁ i)
+        exact Nat.zero_le 2
+    let a : ↑sd := ⟨N + i, hi₁⟩
+    use a
+    constructor
+    . refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_right ↑N ↑i
+    . refine Set.not_mem_Ioo_of_ge ?_
+      have hi₂: ↑↑N + 3 ≤ fd x y a := by
+        refine hi₀ a ?_
+        refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_left ↑i ↑N
+      refine le_trans ?_ hi₂
+      norm_cast
+      exact Nat.le_add_left 1 (↑N + 2)
+
+
+lemma imo_1985_p6_unique_8
+  (f : ℕ → NNReal → ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (x y : NNReal)
+  (hx₀ : ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1)
+  (hy₀ : ∀ (n : ℕ), 0 < n → 0 < f n y ∧ f n y < f (n + 1) y ∧ f (n + 1) y < 1)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₃ : ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1) →
+        (Filter.Tendsto (fd a b) Filter.atTop (nhds 0)))
+  (hy₁ : x < y):
+   Filter.Tendsto (fd x y) Filter.atTop (nhds 0) := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  refine hfd₃ x y hy₁ ?_
+  intro nd
+  have hnd₀: 0 < nd.1 := by
+    refine lt_of_lt_of_le ?_ (hd₁ nd)
+    exact Nat.zero_lt_two
+  have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+  have hnd₂: 0 < nd.1 - 1 := by
+    refine Nat.sub_pos_of_lt ?_
+    refine lt_of_lt_of_le ?_ (hd₁ nd)
+    exact Nat.one_lt_two
+  constructor
+  . constructor
+    . refine h₇ nd.1 x hnd₀ ?_
+      exact (hx₀ (nd.1) hnd₀).2.1
+    . refine h₇ nd.1 y hnd₀ ?_
+      exact (hy₀ (nd.1) hnd₀).2.1
+  . constructor
+    . rw [← hnd₁]
+      exact (hx₀ (nd.1 - 1) hnd₂).2.2
+    . rw [← hnd₁]
+      exact (hy₀ (nd.1 - 1) hnd₂).2.2
+
+lemma imo_1985_p6_unique_9
+  (f : ℕ → NNReal → ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (x y : NNReal)
+  (hx₀ : ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1)
+  (hy₀ : ∀ (n : ℕ), 0 < n → 0 < f n y ∧ f n y < f (n + 1) y ∧ f (n + 1) y < 1)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (nd : ↑sd):
+   (1 - 1 / ↑↑nd < f (↑nd) x ∧ 1 - 1 / ↑↑nd < f (↑nd) y) ∧ f (↑nd) x < 1 ∧ f (↑nd) y < 1 := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hnd₀: 0 < nd.1 := by
+    refine lt_of_lt_of_le ?_ (hd₁ nd)
+    exact Nat.zero_lt_two
+  have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+  have hnd₂: 0 < nd.1 - 1 := by
+    refine Nat.sub_pos_of_lt ?_
+    refine lt_of_lt_of_le ?_ (hd₁ nd)
+    exact Nat.one_lt_two
+  constructor
+  . constructor
+    . refine h₇ nd.1 x hnd₀ ?_
+      exact (hx₀ (nd.1) hnd₀).2.1
+    . refine h₇ nd.1 y hnd₀ ?_
+      exact (hy₀ (nd.1) hnd₀).2.1
+  . constructor
+    . rw [← hnd₁]
+      exact (hx₀ (nd.1 - 1) hnd₂).2.2
+    . rw [← hnd₁]
+      exact (hy₀ (nd.1 - 1) hnd₂).2.2
+
+lemma imo_1985_p6_unique_10
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hy₂ : Filter.Tendsto (fd x y) Filter.atTop Filter.atTop)
+  (hy₃ : Filter.Tendsto (fd x y) Filter.atTop (nhds 0)):
+   False := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hd₂: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    rw [hsd]
+    exact Set.nonempty_Ici
+  apply Filter.tendsto_atTop_atTop.mp at hy₂
+  apply tendsto_atTop_nhds.mp at hy₃
+  contrapose! hy₃
+  clear hy₃
+  let sx : Set ℝ := Set.Ioo (-1) 1
+  use sx
+  constructor
+  . refine Set.mem_Ioo.mpr ?_
+    simp
+  constructor
+  . exact isOpen_Ioo
+  . intro N
+    have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+    obtain ⟨i, hi₀⟩ := hy₅
+    have hi₁: (N.1 + i.1) ∈ sd := by
+      nth_rw 1 [hsd]
+      refine Set.mem_Ici.mpr ?_
+      rw [← add_zero 2]
+      refine Nat.add_le_add ?_ ?_
+      . exact hd₁ N
+      . refine le_trans ?_ (hd₁ i)
+        exact Nat.zero_le 2
+    let a : ↑sd := ⟨N + i, hi₁⟩
+    use a
+    constructor
+    . refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_right ↑N ↑i
+    . refine Set.not_mem_Ioo_of_ge ?_
+      have hi₂: ↑↑N + 3 ≤ fd x y a := by
+        refine hi₀ a ?_
+        refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_left ↑i ↑N
+      refine le_trans ?_ hi₂
+      norm_cast
+      exact Nat.le_add_left 1 (↑N + 2)
+
+
+lemma imo_1985_p6_unique_11
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hy₃ : Filter.Tendsto (fd x y) Filter.atTop (nhds 0))
+  (hy₂ : ∀ (b : ℝ), ∃ i, ∀ (a : ↑sd), i ≤ a → b ≤ fd x y a):
+   False := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hd₂: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    rw [hsd]
+    exact Set.nonempty_Ici
+  apply tendsto_atTop_nhds.mp at hy₃
+  contrapose! hy₃
+  clear hy₃
+  let sx : Set ℝ := Set.Ioo (-1) 1
+  use sx
+  constructor
+  . refine Set.mem_Ioo.mpr ?_
+    simp
+  constructor
+  . exact isOpen_Ioo
+  . intro N
+    have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+    obtain ⟨i, hi₀⟩ := hy₅
+    have hi₁: (N.1 + i.1) ∈ sd := by
+      nth_rw 1 [hsd]
+      refine Set.mem_Ici.mpr ?_
+      rw [← add_zero 2]
+      refine Nat.add_le_add ?_ ?_
+      . exact hd₁ N
+      . refine le_trans ?_ (hd₁ i)
+        exact Nat.zero_le 2
+    let a : ↑sd := ⟨N + i, hi₁⟩
+    use a
+    constructor
+    . refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_right ↑N ↑i
+    . refine Set.not_mem_Ioo_of_ge ?_
+      have hi₂: ↑↑N + 3 ≤ fd x y a := by
+        refine hi₀ a ?_
+        refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_left ↑i ↑N
+      refine le_trans ?_ hi₂
+      norm_cast
+      exact Nat.le_add_left 1 (↑N + 2)
+
+
+lemma imo_1985_p6_unique_12
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hy₂ : ∀ (b : ℝ), ∃ i, ∀ (a : ↑sd), i ≤ a → b ≤ fd x y a)
+  (hy₃ : ∀ (U : Set ℝ), 0 ∈ U → IsOpen U → ∃ N, ∀ (n : ↑sd), N ≤ n → fd x y n ∈ U):
+   False := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  contrapose! hy₃
+  clear hy₃
+  let sx : Set ℝ := Set.Ioo (-1) 1
+  use sx
+  constructor
+  . refine Set.mem_Ioo.mpr ?_
+    simp
+  constructor
+  . exact isOpen_Ioo
+  . intro N
+    have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+    obtain ⟨i, hi₀⟩ := hy₅
+    have hi₁: (N.1 + i.1) ∈ sd := by
+      nth_rw 1 [hsd]
+      refine Set.mem_Ici.mpr ?_
+      rw [← add_zero 2]
+      refine Nat.add_le_add ?_ ?_
+      . exact hd₁ N
+      . refine le_trans ?_ (hd₁ i)
+        exact Nat.zero_le 2
+    let a : ↑sd := ⟨N + i, hi₁⟩
+    use a
+    constructor
+    . refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_right ↑N ↑i
+    . refine Set.not_mem_Ioo_of_ge ?_
+      have hi₂: ↑↑N + 3 ≤ fd x y a := by
+        refine hi₀ a ?_
+        refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_left ↑i ↑N
+      refine le_trans ?_ hi₂
+      norm_cast
+      exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_unique_13
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hy₂ : ∀ (b : ℝ), ∃ i, ∀ (a : ↑sd), i ≤ a → b ≤ fd x y a):
+   ∃ U, 0 ∈ U ∧ IsOpen U ∧ ∀ (N : ↑sd), ∃ n, N ≤ n ∧ fd x y n ∉ U := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  let sx : Set ℝ := Set.Ioo (-1) 1
+  use sx
+  constructor
+  . refine Set.mem_Ioo.mpr ?_
+    simp
+  constructor
+  . exact isOpen_Ioo
+  . intro N
+    have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+    obtain ⟨i, hi₀⟩ := hy₅
+    have hi₁: (N.1 + i.1) ∈ sd := by
+      nth_rw 1 [hsd]
+      refine Set.mem_Ici.mpr ?_
+      rw [← add_zero 2]
+      refine Nat.add_le_add ?_ ?_
+      . exact hd₁ N
+      . refine le_trans ?_ (hd₁ i)
+        exact Nat.zero_le 2
+    let a : ↑sd := ⟨N + i, hi₁⟩
+    use a
+    constructor
+    . refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_right ↑N ↑i
+    . refine Set.not_mem_Ioo_of_ge ?_
+      have hi₂: ↑↑N + 3 ≤ fd x y a := by
+        refine hi₀ a ?_
+        refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_left ↑i ↑N
+      refine le_trans ?_ hi₂
+      norm_cast
+      exact Nat.le_add_left 1 (↑N + 2)
+
+
+lemma imo_1985_p6_unique_14
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hy₂ : ∀ (b : ℝ), ∃ i, ∀ (a : ↑sd), i ≤ a → b ≤ fd x y a)
+  (sx : Set ℝ)
+  (hsx : sx = Set.Ioo (-1) 1):
+   ∃ U, 0 ∈ U ∧ IsOpen U ∧ ∀ (N : ↑sd), ∃ n, N ≤ n ∧ fd x y n ∉ U := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  use sx
+  constructor
+  . rw [hsx]
+    refine Set.mem_Ioo.mpr ?_
+    simp
+  constructor
+  . rw [hsx]
+    exact isOpen_Ioo
+  . intro N
+    have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+    obtain ⟨i, hi₀⟩ := hy₅
+    have hi₁: (N.1 + i.1) ∈ sd := by
+      nth_rw 1 [hsd]
+      refine Set.mem_Ici.mpr ?_
+      rw [← add_zero 2]
+      refine Nat.add_le_add ?_ ?_
+      . exact hd₁ N
+      . refine le_trans ?_ (hd₁ i)
+        exact Nat.zero_le 2
+    let a : ↑sd := ⟨N + i, hi₁⟩
+    use a
+    constructor
+    . refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_right ↑N ↑i
+    . rw [hsx]
+      refine Set.not_mem_Ioo_of_ge ?_
+      have hi₂: ↑↑N + 3 ≤ fd x y a := by
+        refine hi₀ a ?_
+        refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_left ↑i ↑N
+      refine le_trans ?_ hi₂
+      norm_cast
+      exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_unique_15
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hy₂ : ∀ (b : ℝ), ∃ i, ∀ (a : ↑sd), i ≤ a → b ≤ fd x y a)
+  (sx : Set ℝ)
+  (hsx : sx = Set.Ioo (-1) 1):
+   0 ∈ sx ∧ IsOpen sx ∧ ∀ (N : ↑sd), ∃ n, N ≤ n ∧ fd x y n ∉ sx := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  constructor
+  . rw [hsx]
+    refine Set.mem_Ioo.mpr ?_
+    simp
+  constructor
+  . rw [hsx]
+    exact isOpen_Ioo
+  . intro N
+    have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+    obtain ⟨i, hi₀⟩ := hy₅
+    have hi₁: (N.1 + i.1) ∈ sd := by
+      nth_rw 1 [hsd]
+      refine Set.mem_Ici.mpr ?_
+      rw [← add_zero 2]
+      refine Nat.add_le_add ?_ ?_
+      . exact hd₁ N
+      . refine le_trans ?_ (hd₁ i)
+        exact Nat.zero_le 2
+    let a : ↑sd := ⟨N + i, hi₁⟩
+    use a
+    constructor
+    . refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_right ↑N ↑i
+    . rw [hsx]
+      refine Set.not_mem_Ioo_of_ge ?_
+      have hi₂: ↑↑N + 3 ≤ fd x y a := by
+        refine hi₀ a ?_
+        refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_left ↑i ↑N
+      refine le_trans ?_ hi₂
+      norm_cast
+      exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_unique_16
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hy₂ : ∀ (b : ℝ), ∃ i, ∀ (a : ↑sd), i ≤ a → b ≤ fd x y a)
+  (sx : Set ℝ)
+  (hsx : sx = Set.Ioo (-1) 1):
+   IsOpen sx ∧ ∀ (N : ↑sd), ∃ n, N ≤ n ∧ fd x y n ∉ sx := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  constructor
+  . rw [hsx]
+    exact isOpen_Ioo
+  . intro N
+    have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+    obtain ⟨i, hi₀⟩ := hy₅
+    have hi₁: (N.1 + i.1) ∈ sd := by
+      nth_rw 1 [hsd]
+      refine Set.mem_Ici.mpr ?_
+      rw [← add_zero 2]
+      refine Nat.add_le_add ?_ ?_
+      . exact hd₁ N
+      . refine le_trans ?_ (hd₁ i)
+        exact Nat.zero_le 2
+    let a : ↑sd := ⟨N + i, hi₁⟩
+    use a
+    constructor
+    . refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_right ↑N ↑i
+    . rw [hsx]
+      refine Set.not_mem_Ioo_of_ge ?_
+      have hi₂: ↑↑N + 3 ≤ fd x y a := by
+        refine hi₀ a ?_
+        refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_left ↑i ↑N
+      refine le_trans ?_ hi₂
+      norm_cast
+      exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_unique_17
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hy₂ : ∀ (b : ℝ), ∃ i, ∀ (a : ↑sd), i ≤ a → b ≤ fd x y a)
+  (sx : Set ℝ)
+  (hsx : sx = Set.Ioo (-1) 1)
+  (N : ↑sd):
+   ∃ n, N ≤ n ∧ fd x y n ∉ sx := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+  obtain ⟨i, hi₀⟩ := hy₅
+  have hi₁: (N.1 + i.1) ∈ sd := by
+    nth_rw 1 [hsd]
+    refine Set.mem_Ici.mpr ?_
+    rw [← add_zero 2]
+    refine Nat.add_le_add ?_ ?_
+    . exact hd₁ N
+    . refine le_trans ?_ (hd₁ i)
+      exact Nat.zero_le 2
+  let a : ↑sd := ⟨N + i, hi₁⟩
+  use a
+  constructor
+  . refine Subtype.mk_le_mk.mpr ?_
+    exact Nat.le_add_right ↑N ↑i
+  . rw [hsx]
+    refine Set.not_mem_Ioo_of_ge ?_
+    have hi₂: ↑↑N + 3 ≤ fd x y a := by
+      refine hi₀ a ?_
+      refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_left ↑i ↑N
+    refine le_trans ?_ hi₂
+    norm_cast
+    exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_unique_18
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (sx : Set ℝ)
+  (hsx : sx = Set.Ioo (-1) 1)
+  (N i : ↑sd)
+  (hi₀ : ∀ (a : ↑sd), i ≤ a → ↑↑N + 3 ≤ fd x y a):
+   ∃ n, N ≤ n ∧ fd x y n ∉ sx := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  have hi₁: (N.1 + i.1) ∈ sd := by
+    nth_rw 1 [hsd]
+    refine Set.mem_Ici.mpr ?_
+    rw [← add_zero 2]
+    refine Nat.add_le_add ?_ ?_
+    . exact hd₁ N
+    . refine le_trans ?_ (hd₁ i)
+      exact Nat.zero_le 2
+  let a : ↑sd := ⟨N + i, hi₁⟩
+  use a
+  constructor
+  . refine Subtype.mk_le_mk.mpr ?_
+    exact Nat.le_add_right ↑N ↑i
+  . rw [hsx]
+    refine Set.not_mem_Ioo_of_ge ?_
+    have hi₂: ↑↑N + 3 ≤ fd x y a := by
+      refine hi₀ a ?_
+      refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_left ↑i ↑N
+    refine le_trans ?_ hi₂
+    norm_cast
+    exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_unique_19
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (N i : ↑sd):
+   N.1 + ↑i ∈ sd := by
+  have hd₁: ∀ nd:↑sd, 2 ≤ nd.1 := by
+    intro nd
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  nth_rw 1 [hsd]
+  refine Set.mem_Ici.mpr ?_
+  rw [← add_zero 2]
+  refine Nat.add_le_add ?_ ?_
+  . exact hd₁ N
+  . refine le_trans ?_ (hd₁ i)
+    exact Nat.zero_le 2
+
+lemma imo_1985_p6_unique_20
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (sx : Set ℝ)
+  (hsx : sx = Set.Ioo (-1) 1)
+  (N i : ↑sd)
+  (hi₀ : ∀ (a : ↑sd), i ≤ a → ↑↑N + 3 ≤ fd x y a)
+  (hi₁ : N.1 + ↑i ∈ sd):
+   ∃ n, N ≤ n ∧ fd x y n ∉ sx := by
+  let a : ↑sd := ⟨N + i, hi₁⟩
+  use a
+  constructor
+  . refine Subtype.mk_le_mk.mpr ?_
+    exact Nat.le_add_right ↑N ↑i
+  . rw [hsx]
+    refine Set.not_mem_Ioo_of_ge ?_
+    have hi₂: ↑↑N + 3 ≤ fd x y a := by
+      refine hi₀ a ?_
+      refine Subtype.mk_le_mk.mpr ?_
+      exact Nat.le_add_left ↑i ↑N
+    refine le_trans ?_ hi₂
+    norm_cast
+    exact Nat.le_add_left 1 (↑N + 2)
+
+lemma imo_1985_p6_unique_21
+  (x y : NNReal)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (sx : Set ℝ)
+  (hsx : sx = Set.Ioo (-1) 1)
+  (N i : ↑sd)
+  (hi₀ : ∀ (a : ↑sd), i ≤ a → ↑↑N + 3 ≤ fd x y a)
+  (hi₁ : N.1 + ↑i ∈ sd)
+  (a : ↑sd)
+  (ha : a = ⟨↑N + ↑i, hi₁⟩):
+   ∃ n, N ≤ n ∧ fd x y n ∉ sx := by
+  use a
+  constructor
+  . refine Subtype.mk_le_mk.mpr ?_
+    rw [ha]
+    exact Nat.le_add_right ↑N ↑i
+  . rw [hsx]
+    refine Set.not_mem_Ioo_of_ge ?_
+    have hi₂: ↑↑N + 3 ≤ fd x y a := by
+      refine hi₀ a ?_
+      refine Subtype.mk_le_mk.mpr ?_
+      rw [ha]
+      exact Nat.le_add_left ↑i ↑N
+    refine le_trans ?_ hi₂
+    norm_cast
+    exact Nat.le_add_left 1 (↑N + 2)
+
+
+lemma imo_1985_p6_exists_27
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (nn : ↑sn) :
+   ↑(fb nn) ∈ fr '' sb := by
+  refine (Set.mem_image fr sb _).mpr ?_
+  use (fb nn)
+  rw [hfr, hsb₀]
+  constructor
+  . exact Set.mem_range_self nn
+  . exact rfl
+
+lemma imo_1985_p6_exists_28
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (nn : ↑sn) :
+   ∃ x ∈ sb, fr x = ↑(fb nn) := by
+  use (fb nn)
+  rw [hfr, hsb₀]
+  constructor
+  . exact Set.mem_range_self nn
+  . exact rfl
+
+lemma imo_1985_p6_exists_29
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n, hn₁⟩)
+  (hbr₅ : ↑(fb nn) = br):
+   False := by
+  have hn₂: n + 1 ∈ sn := by
+    rw [hsn₀]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  let ns : ↑sn := ⟨n + 1, hn₂⟩
+  have hc₁: fb nn < fb ns := by
+    refine hfb₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hnn]
+    exact lt_add_one n
+  have hbr₆: fb ns ≤ fb nn := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hbr₅]
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    use (fb ns)
+    rw [hfr, hsb₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fb nn)).mp ?_
+  exact lt_of_lt_of_le hc₁ hbr₆
+
+lemma imo_1985_p6_exists_30
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (n : ℕ)
+  (hn₁ : n ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n, hn₁⟩)
+  (hn₂ : n + 1 ∈ sn)
+  (ns : ↑sn)
+  (hns : ns = ⟨n + 1, hn₂⟩):
+   fb nn < fb ns := by
+  refine hfb₃ ?_
+  rw [hnn, hns]
+  refine Subtype.mk_lt_mk.mpr ?_
+  exact lt_add_one n
+
+
+lemma imo_1985_p6_exists_31
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (nn : ↑sn)
+  (hbr₅ : ↑(fb nn) = br)
+  (ns : ↑sn)
+  (hc₁ : fb nn < fb ns):
+   False := by
+  have hbr₆: fb ns ≤ fb nn := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hbr₅]
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    use (fb ns)
+    rw [hfr, hsb₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fb nn)).mp ?_
+  exact lt_of_lt_of_le hc₁ hbr₆
+
+lemma imo_1985_p6_exists_32
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (nn : ↑sn)
+  (hbr₅ : ↑(fb nn) = br)
+  (ns : ↑sn):
+   fb ns ≤ fb nn := by
+  refine NNReal.coe_le_coe.mp ?_
+  rw [hbr₅]
+  refine hbr₃ _ ?_
+  rw [hsbr]
+  refine (Set.mem_image fr sb _).mpr ?_
+  use (fb ns)
+  rw [hfr, hsb₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self ns
+
+lemma imo_1985_p6_exists_33
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (br : ℝ)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n, hn₁⟩)
+  (hn₂ : ↑(fb nn) < br):
+   1 - 1 / ↑n < f n br.toNNReal := by
+  have hn₃: f n (fb nn) = 1 - 1 / n := by
+    rw [hf₁ n _ hn₀, hnn, hfb₁ ⟨n, hn₁⟩]
+    refine NNReal.coe_sub ?_
+    refine div_le_self ?_ ?_
+    . exact zero_le_one' NNReal
+    . exact Nat.one_le_cast.mpr hn₀
+  rw [← hn₃]
+  refine hmo₀ n hn₀ ?_
+  exact Real.lt_toNNReal_iff_coe_lt.mpr hn₂
+
+lemma imo_1985_p6_exists_34
+  (f : ℕ → NNReal → ℝ)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n, hn₁⟩):
+   f n (fb nn) = 1 - 1 / ↑n := by
+  rw [hf₁ n _ hn₀, hnn, hfb₁ ⟨n, hn₁⟩]
+  refine NNReal.coe_sub ?_
+  refine div_le_self ?_ ?_
+  . exact zero_le_one' NNReal
+  . exact Nat.one_le_cast.mpr hn₀
+
+lemma imo_1985_p6_exists_35
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (br : ℝ)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (nn : ↑sn)
+  (hn₂ : ↑(fb nn) < br)
+  (hn₃ : f n (fb nn) = 1 - 1 / ↑n):
+   1 - 1 / ↑n < f n br.toNNReal := by
+  rw [← hn₃]
+  refine hmo₀ n hn₀ ?_
+  exact Real.lt_toNNReal_iff_coe_lt.mpr hn₂
+
+
+lemma imo_1985_p6_exists_36
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (hbr₁ : 0 < br)
+  (hu₅ : br ≤ cr)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (hu₆ : br = cr)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+  f (n + 1) br.toNNReal < 1 := by
+  have hn₂: n + 1 ∈ sn := by
+    rw [hsn₀]
+    exact Set.mem_Ici.mpr (by linarith)
+  let nn : ↑sn := ⟨n + 1, hn₂⟩
+  have hcr₁: 0 < cr := by exact gt_of_ge_of_gt hu₅ hbr₁
+  have hn₃: f (n + 1) (fc (nn)) = 1 := by
+    rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+    exact rfl
+  rw [← hn₃, hu₆]
+  refine hmo₀ (n + 1) (by linarith) ?_
+  refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt hcr₁)).mpr ?_
+  by_contra! hc₀
+  have hc₁: fc nn = cr := by
+    refine eq_of_le_of_le hc₀ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+  have hn₄: n + 2 ∈ sn := by
+    rw [hsn₀]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  let ns : ↑sn := ⟨n + 2, hn₄⟩
+  have hn₅: fc ns < fc nn := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    exact Nat.lt_add_one (n + 1)
+  have hc₂: fc nn ≤ fc ns := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hc₁]
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc ns)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fc ns)).mp ?_
+  exact lt_of_lt_of_le hn₅ hc₂
+
+lemma imo_1985_p6_exists_37
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (hu₆ : br = cr)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n + 1, hn₂⟩)
+  (hcr₁ : 0 < cr)
+  (hn₃ : f (n + 1) (fc nn) = 1):
+   f (n + 1) br.toNNReal < 1 := by
+  rw [← hn₃, hu₆]
+  refine hmo₀ (n + 1) (by linarith) ?_
+  refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt hcr₁)).mpr ?_
+  by_contra! hc₀
+  have hc₁: fc nn = cr := by
+    refine eq_of_le_of_le hc₀ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+  have hn₄: n + 2 ∈ sn := by
+    rw [hsn₀]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  let ns : ↑sn := ⟨n + 2, hn₄⟩
+  have hn₅: fc ns < fc nn := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hnn]
+    exact Nat.lt_add_one (n + 1)
+  have hc₂: fc nn ≤ fc ns := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hc₁]
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc ns)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fc ns)).mp ?_
+  exact lt_of_lt_of_le hn₅ hc₂
+
+
+lemma imo_1985_p6_exists_38
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n + 1, hn₂⟩)
+  (hcr₁ : 0 < cr):
+   f (n + 1) cr.toNNReal < f (n + 1) (fc nn) := by
+  refine hmo₀ (n + 1) (by linarith) ?_
+  refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt hcr₁)).mpr ?_
+  by_contra! hc₀
+  have hc₁: fc nn = cr := by
+    refine eq_of_le_of_le hc₀ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+  have hn₄: n + 2 ∈ sn := by
+    rw [hsn₀]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  let ns : ↑sn := ⟨n + 2, hn₄⟩
+  have hn₅: fc ns < fc nn := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hnn]
+    exact Nat.lt_add_one (n + 1)
+  have hc₂: fc nn ≤ fc ns := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hc₁]
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc ns)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fc ns)).mp ?_
+  exact lt_of_lt_of_le hn₅ hc₂
+
+lemma imo_1985_p6_exists_39
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n + 1, hn₂⟩)
+  (hcr₁ : 0 < cr):
+   cr.toNNReal < fc nn := by
+  refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt hcr₁)).mpr ?_
+  by_contra! hc₀
+  have hc₁: fc nn = cr := by
+    refine eq_of_le_of_le hc₀ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+  have hn₄: n + 2 ∈ sn := by
+    rw [hsn₀]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  let ns : ↑sn := ⟨n + 2, hn₄⟩
+  have hn₅: fc ns < fc nn := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hnn]
+    exact Nat.lt_add_one (n + 1)
+  have hc₂: fc nn ≤ fc ns := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hc₁]
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc ns)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fc ns)).mp ?_
+  exact lt_of_lt_of_le hn₅ hc₂
+
+lemma imo_1985_p6_exists_40
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n + 1, hn₂⟩):
+   cr < ↑(fc nn) := by
+  by_contra! hc₀
+  have hc₁: fc nn = cr := by
+    refine eq_of_le_of_le hc₀ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+  have hn₄: n + 2 ∈ sn := by
+    rw [hsn₀]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  let ns : ↑sn := ⟨n + 2, hn₄⟩
+  have hn₅: fc ns < fc nn := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hnn]
+    exact Nat.lt_add_one (n + 1)
+  have hc₂: fc nn ≤ fc ns := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hc₁]
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc ns)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fc ns)).mp ?_
+  exact lt_of_lt_of_le hn₅ hc₂
+
+
+
+lemma imo_1985_p6_exists_41
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n + 1, hn₂⟩)
+  (hc₀ : ↑(fc nn) ≤ cr):
+   False := by
+  have hc₁: fc nn = cr := by
+    refine eq_of_le_of_le hc₀ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+  have hn₄: n + 2 ∈ sn := by
+    rw [hsn₀]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  let ns : ↑sn := ⟨n + 2, hn₄⟩
+  have hn₅: fc ns < fc nn := by
+    refine hfc₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hnn]
+    exact Nat.lt_add_one (n + 1)
+  have hc₂: fc nn ≤ fc ns := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hc₁]
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc ns)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fc ns)).mp ?_
+  exact lt_of_lt_of_le hn₅ hc₂
+
+
+lemma imo_1985_p6_exists_42
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (nn : ↑sn)
+  (hc₀ : ↑(fc nn) ≤ cr):
+   ↑(fc nn) = cr := by
+  refine eq_of_le_of_le hc₀ ?_
+  refine hcr₃ _ ?_
+  rw [hscr]
+  refine (Set.mem_image fr sc _).mpr ?_
+  use (fc nn)
+  rw [hfr, hsc₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_43
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+  n + 2 ∈ sn := by
+  rw [hsn₀]
+  refine Set.mem_Ici.mpr ?_
+  exact Nat.le_add_right_of_le hn₀
+
+lemma imo_1985_p6_exists_44
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (hfc₃ : StrictAnti fc)
+  (n : ℕ)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n + 1, hn₂⟩)
+  (hn₄ : n + 2 ∈ sn)
+  (ns : ↑sn)
+  (hns : ns = ⟨n + 2, hn₄⟩):
+  fc ns < fc nn := by
+  refine hfc₃ ?_
+  rw [hnn, hns]
+  refine Subtype.mk_lt_mk.mpr ?_
+  exact Nat.lt_add_one (n + 1)
+
+
+lemma imo_1985_p6_exists_45
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (n : ℕ)
+  (nn : ↑sn)
+  (hc₁ : ↑(fc nn) = cr)
+  (hn₄ : n + 2 ∈ sn)
+  (ns : ↑sn := ⟨n + 2, hn₄⟩)
+  (hn₅ : fc ns < fc nn):
+   False := by
+  have hc₂: fc nn ≤ fc ns := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hc₁]
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc ns)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fc ns)).mp ?_
+  exact lt_of_lt_of_le hn₅ hc₂
+
+
+lemma imo_1985_p6_exists_46
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (n : ℕ)
+  (nn : ↑sn)
+  (hc₁ : ↑(fc nn) = cr)
+  (hn₄ : n + 2 ∈ sn)
+  (ns : ↑sn := ⟨n + 2, hn₄⟩):
+   fc nn ≤ fc ns := by
+  refine NNReal.coe_le_coe.mp ?_
+  rw [hc₁]
+  refine hcr₃ _ ?_
+  rw [hscr]
+  refine (Set.mem_image fr sc _).mpr ?_
+  use (fc ns)
+  rw [hfr, hsc₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self ns
+
+
+lemma imo_1985_p6_unique_top_ind_1
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hd₃ : ∀ (nd : ↑sd), nd.1 + 1 ∈ sd)
+  (hd₂ : ∀ (nd : ↑sd), fd a b nd * (2 - 1 / ↑↑nd) ≤ fd a b ⟨nd.1 + 1, hd₃ nd⟩)
+  (hi₀ : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi₀⟩)
+  (nd : ↑sd)
+  (hnd₀ : 2 ≤ nd.1):
+   fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ f (↑nd) b - f (↑nd) a := by
+  refine Nat.le_induction ?_ ?_ nd.1 hnd₀
+  . have hi₂: i.val = (2:ℕ) := by
+      simp_all only [Subtype.forall]
+    rw [hfd₁ a b i, hi₂]
+    simp
+  . simp
+    intros n hn₀ hn₁
+    have hn₂: n - 1 = n - 2 + 1 := by
+      simp
+      exact (Nat.sub_eq_iff_eq_add hn₀).mp rfl
+    have hn₃: n ∈ sd := by
+      rw [hsd]
+      exact hn₀
+    let nn : ↑sd := ⟨n, hn₃⟩
+    have hnn: nn.1 = n := by exact rfl
+    have hn₄: nn.1 + 1 ∈ sd := by
+      rw [hnn, hsd]
+      refine Set.mem_Ici.mpr ?_
+      exact Nat.le_add_right_of_le hn₀
+    have hn₅: fd a b nn * (2 - 1 / ↑n) ≤ fd a b ⟨nn.1 + 1, hn₄⟩ := by exact hd₂ nn
+    rw [hfd₁ a b ⟨nn.1 + 1, hn₄⟩] at hn₅
+    have hn₆: f (↑nn + 1) b - f (↑nn + 1) a = f (n + 1) b - f (n + 1) a := by exact rfl
+    rw [hn₆] at hn₅
+    refine le_trans ?_ hn₅
+    rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+    refine mul_le_mul ?_ ?_ (by linarith) ?_
+    . refine le_of_le_of_eq hn₁ ?_
+      rw [hfd₁]
+    . refine (div_le_iff₀ (two_pos)).mpr ?_
+      rw [sub_mul, one_div_mul_eq_div _ 2]
+      refine le_sub_iff_add_le.mpr ?_
+      refine le_sub_iff_add_le'.mp ?_
+      refine (div_le_iff₀ ?_).mpr ?_
+      . refine Nat.cast_pos.mpr ?_
+        exact lt_of_lt_of_le (two_pos) hn₀
+      . ring_nf
+        exact Nat.ofNat_le_cast.mpr hn₀
+    . exact le_of_lt (hd₁ nn a b ha₀)
+
+
+
+
+
+
+lemma imo_1985_p6_unique_top_ind_2
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (a b : NNReal)
+  (hi₀ : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi₀⟩):
+   fd a b i * (3 / 2) ^ (2 - 2) ≤ f 2 b - f 2 a := by
+  have hi₂: i.val = (2:ℕ) := by
+    simp_all only [Subtype.forall]
+  rw [hfd₁ a b i, hi₂]
+  simp
+
+
+lemma imo_1985_p6_unique_top_ind_3
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hd₃ : ∀ (nd : ↑sd), nd.1 + 1 ∈ sd)
+  (hd₂ : ∀ (nd : ↑sd), fd a b nd * (2 - 1 / ↑↑nd) ≤ fd a b ⟨nd.1 + 1, hd₃ nd⟩)
+  (i : ↑sd):
+   ∀ (n : ℕ),
+    2 ≤ n → fd a b i * (3 / 2) ^ (n - 2) ≤ f n b - f n a → fd a b i * (3 / 2) ^ (n + 1 - 2) ≤ f (n + 1) b - f (n + 1) a := by
+  simp
+  intros n hn₀ hn₁
+  have hn₂: n - 1 = n - 2 + 1 := by
+    simp
+    exact (Nat.sub_eq_iff_eq_add hn₀).mp rfl
+  have hn₃: n ∈ sd := by
+    rw [hsd]
+    exact hn₀
+  let nn : ↑sd := ⟨n, hn₃⟩
+  have hnn: nn.1 = n := by exact rfl
+  have hn₄: nn.1 + 1 ∈ sd := by
+    rw [hnn, hsd]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  have hn₅: fd a b nn * (2 - 1 / ↑n) ≤ fd a b ⟨nn.1 + 1, hn₄⟩ := by exact hd₂ nn
+  rw [hfd₁ a b ⟨nn.1 + 1, hn₄⟩] at hn₅
+  have hn₆: f (↑nn + 1) b - f (↑nn + 1) a = f (n + 1) b - f (n + 1) a := by exact rfl
+  rw [hn₆] at hn₅
+  refine le_trans ?_ hn₅
+  rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+  refine mul_le_mul ?_ ?_ (by linarith) ?_
+  . refine le_of_le_of_eq hn₁ ?_
+    rw [hfd₁]
+  . refine (div_le_iff₀ (two_pos)).mpr ?_
+    rw [sub_mul, one_div_mul_eq_div _ 2]
+    refine le_sub_iff_add_le.mpr ?_
+    refine le_sub_iff_add_le'.mp ?_
+    refine (div_le_iff₀ ?_).mpr ?_
+    . refine Nat.cast_pos.mpr ?_
+      exact lt_of_lt_of_le (two_pos) hn₀
+    . ring_nf
+      exact Nat.ofNat_le_cast.mpr hn₀
+  . exact le_of_lt (hd₁ nn a b ha₀)
+
+lemma imo_1985_p6_unique_top_ind_4
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hd₃ : ∀ (nd : ↑sd), nd.1 + 1 ∈ sd)
+  (hd₂ : ∀ (nd : ↑sd), fd a b nd * (2 - 1 / ↑↑nd) ≤ fd a b ⟨nd.1 + 1, hd₃ nd⟩)
+  (i : ↑sd)
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (hn₁ : fd a b i * (3 / 2) ^ (n - 2) ≤ f n b - f n a)
+  (hn₂ : n - 1 = n - 2 + 1):
+   fd a b i * (3 / 2) ^ (n - 1) ≤ f (n + 1) b - f (n + 1) a := by
+  have hn₃: n ∈ sd := by
+    rw [hsd]
+    exact hn₀
+  let nn : ↑sd := ⟨n, hn₃⟩
+  have hnn: nn.1 = n := by exact rfl
+  have hn₄: nn.1 + 1 ∈ sd := by
+    rw [hnn, hsd]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  have hn₅: fd a b nn * (2 - 1 / ↑n) ≤ fd a b ⟨nn.1 + 1, hn₄⟩ := by exact hd₂ nn
+  rw [hfd₁ a b ⟨nn.1 + 1, hn₄⟩] at hn₅
+  have hn₆: f (↑nn + 1) b - f (↑nn + 1) a = f (n + 1) b - f (n + 1) a := by exact rfl
+  rw [hn₆] at hn₅
+  refine le_trans ?_ hn₅
+  rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+  refine mul_le_mul ?_ ?_ (by linarith) ?_
+  . refine le_of_le_of_eq hn₁ ?_
+    rw [hfd₁]
+  . refine (div_le_iff₀ (two_pos)).mpr ?_
+    rw [sub_mul, one_div_mul_eq_div _ 2]
+    refine le_sub_iff_add_le.mpr ?_
+    refine le_sub_iff_add_le'.mp ?_
+    refine (div_le_iff₀ ?_).mpr ?_
+    . refine Nat.cast_pos.mpr ?_
+      exact lt_of_lt_of_le (two_pos) hn₀
+    . ring_nf
+      exact Nat.ofNat_le_cast.mpr hn₀
+  . exact le_of_lt (hd₁ nn a b ha₀)
+
+lemma imo_1985_p6_unique_top_ind_5
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hd₃ : ∀ (nd : ↑sd), nd.1 + 1 ∈ sd)
+  (hd₂ : ∀ (nd : ↑sd), fd a b nd * (2 - 1 / ↑↑nd) ≤ fd a b ⟨nd.1 + 1, hd₃ nd⟩)
+  (i : ↑sd)
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (hn₁ : fd a b i * (3 / 2) ^ (n - 2) ≤ f n b - f n a)
+  (hn₂ : n - 1 = n - 2 + 1)
+  (hn₃ : n ∈ sd)
+  (nn : ↑sd := ⟨n, hn₃⟩)
+  (hnn : nn.1 = n)
+  (hn₄ : nn.1 + 1 ∈ sd) :
+  fd a b i * (3 / 2) ^ (n - 1) ≤ f (n + 1) b - f (n + 1) a := by
+  have hn₅: fd a b nn * (2 - 1 / ↑n) ≤ fd a b ⟨nn.1 + 1, hn₄⟩ := by
+    rw [← hnn]
+    exact hd₂ nn
+  rw [hfd₁ a b ⟨nn.1 + 1, hn₄⟩] at hn₅
+  have hn₆: f (↑nn + 1) b - f (↑nn + 1) a = f (n + 1) b - f (n + 1) a := by rw [hnn]
+  rw [hn₆] at hn₅
+  refine le_trans ?_ hn₅
+  rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+  refine mul_le_mul ?_ ?_ (by linarith) ?_
+  . refine le_of_le_of_eq hn₁ ?_
+    rw [hfd₁, hnn]
+  . refine (div_le_iff₀ (two_pos)).mpr ?_
+    rw [sub_mul, one_div_mul_eq_div _ 2]
+    refine le_sub_iff_add_le.mpr ?_
+    refine le_sub_iff_add_le'.mp ?_
+    refine (div_le_iff₀ ?_).mpr ?_
+    . refine Nat.cast_pos.mpr ?_
+      exact lt_of_lt_of_le (two_pos) hn₀
+    . ring_nf
+      exact Nat.ofNat_le_cast.mpr hn₀
+  . exact le_of_lt (hd₁ nn a b ha₀)
+
+
+lemma imo_1985_p6_unique_top_ind_6
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (i : ↑sd)
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (hn₁ : fd a b i * (3 / 2) ^ (n - 2) ≤ f n b - f n a)
+  (hn₂ : n - 1 = n - 2 + 1)
+  (hn₃ : n ∈ sd)
+  (nn : ↑sd := ⟨n, hn₃⟩)
+  (hnn : ↑nn = n)
+  (hn₄ : nn.1 + 1 ∈ sd)
+  (hn₅ : fd a b nn * (2 - 1 / ↑n) ≤ fd a b ⟨↑nn + 1, hn₄⟩):
+   fd a b i * (3 / 2) ^ (n - 1) ≤ f (n + 1) b - f (n + 1) a := by
+  rw [hfd₁ a b ⟨nn.1 + 1, hn₄⟩] at hn₅
+  have hn₆: f (↑nn + 1) b - f (↑nn + 1) a = f (n + 1) b - f (n + 1) a := by rw [hnn]
+  rw [hn₆] at hn₅
+  refine le_trans ?_ hn₅
+  rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+  refine mul_le_mul ?_ ?_ (by linarith) ?_
+  . refine le_of_le_of_eq hn₁ ?_
+    rw [hfd₁, hnn]
+  . refine (div_le_iff₀ (two_pos)).mpr ?_
+    rw [sub_mul, one_div_mul_eq_div _ 2]
+    refine le_sub_iff_add_le.mpr ?_
+    refine le_sub_iff_add_le'.mp ?_
+    refine (div_le_iff₀ ?_).mpr ?_
+    . refine Nat.cast_pos.mpr ?_
+      exact lt_of_lt_of_le (two_pos) hn₀
+    . ring_nf
+      exact Nat.ofNat_le_cast.mpr hn₀
+  . exact le_of_lt (hd₁ nn a b ha₀)
+
+lemma imo_1985_p6_unique_top_ind_7
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (i : ↑sd)
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (hn₁ : fd a b i * (3 / 2) ^ (n - 2) ≤ f n b - f n a)
+  (hn₂ : n - 1 = n - 2 + 1)
+  (hn₃ : n ∈ sd)
+  (nn : ↑sd := ⟨n, hn₃⟩)
+  (hnn : ↑nn = n)
+  (hn₅ : fd a b nn * (2 - 1 / ↑n) ≤ f (n + 1) b - f (n + 1) a):
+   fd a b i * (3 / 2) ^ (n - 1) ≤ f (n + 1) b - f (n + 1) a := by
+  refine le_trans ?_ hn₅
+  rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+  refine mul_le_mul ?_ ?_ (by linarith) ?_
+  . refine le_of_le_of_eq hn₁ ?_
+    rw [hfd₁, hnn]
+  . refine (div_le_iff₀ (two_pos)).mpr ?_
+    rw [sub_mul, one_div_mul_eq_div _ 2]
+    refine le_sub_iff_add_le.mpr ?_
+    refine le_sub_iff_add_le'.mp ?_
+    refine (div_le_iff₀ ?_).mpr ?_
+    . refine Nat.cast_pos.mpr ?_
+      exact lt_of_lt_of_le (two_pos) hn₀
+    . ring_nf
+      exact Nat.ofNat_le_cast.mpr hn₀
+  . exact le_of_lt (hd₁ nn a b ha₀)
+
+lemma imo_1985_p6_unique_top_ind_8
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (i : ↑sd)
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (hn₁ : fd a b i * (3 / 2) ^ (n - 2) ≤ f n b - f n a)
+  (hn₂ : n - 1 = n - 2 + 1)
+  (hn₃ : n ∈ sd)
+  (nn : ↑sd := ⟨n, hn₃⟩)
+  (hnn : ↑nn = n):
+   fd a b i * (3 / 2) ^ (n - 1) ≤ fd a b nn * (2 - 1 / ↑n) := by
+  rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+  refine mul_le_mul ?_ ?_ (by linarith) ?_
+  . refine le_of_le_of_eq hn₁ ?_
+    rw [hfd₁, hnn]
+  . refine (div_le_iff₀ (two_pos)).mpr ?_
+    rw [sub_mul, one_div_mul_eq_div _ 2]
+    refine le_sub_iff_add_le.mpr ?_
+    refine le_sub_iff_add_le'.mp ?_
+    refine (div_le_iff₀ ?_).mpr ?_
+    . refine Nat.cast_pos.mpr ?_
+      exact lt_of_lt_of_le (two_pos) hn₀
+    . ring_nf
+      exact Nat.ofNat_le_cast.mpr hn₀
+  . exact le_of_lt (hd₁ nn a b ha₀)
+
+lemma imo_1985_p6_unique_top_ind_9
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (i : ↑sd)
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (hn₁ : fd a b i * (3 / 2) ^ (n - 2) ≤ f n b - f n a)
+  (hn₃ : n ∈ sd)
+  (nn : ↑sd := ⟨n, hn₃⟩)
+  (hnn : ↑nn = n):
+   fd a b i * (3 / 2) ^ (n - 2) * (3 / 2) ≤ fd a b nn * (2 - 1 / ↑n) := by
+  refine mul_le_mul ?_ ?_ (by linarith) ?_
+  . refine le_of_le_of_eq hn₁ ?_
+    rw [hfd₁, hnn]
+  . refine (div_le_iff₀ (two_pos)).mpr ?_
+    rw [sub_mul, one_div_mul_eq_div _ 2]
+    refine le_sub_iff_add_le.mpr ?_
+    refine le_sub_iff_add_le'.mp ?_
+    refine (div_le_iff₀ ?_).mpr ?_
+    . refine Nat.cast_pos.mpr ?_
+      exact lt_of_lt_of_le (two_pos) hn₀
+    . ring_nf
+      exact Nat.ofNat_le_cast.mpr hn₀
+  . exact le_of_lt (hd₁ nn a b ha₀)
+
+lemma imo_1985_p6_unique_top_ind_10
+  (n : ℕ)
+  (hn₀ : 2 ≤ n):
+  (3:ℝ) / 2 ≤ 2 - 1 / ↑n := by
+  refine (div_le_iff₀ (two_pos)).mpr ?_
+  rw [sub_mul, one_div_mul_eq_div _ 2]
+  refine le_sub_iff_add_le.mpr ?_
+  refine le_sub_iff_add_le'.mp ?_
+  refine (div_le_iff₀ ?_).mpr ?_
+  . refine Nat.cast_pos.mpr ?_
+    exact lt_of_lt_of_le (two_pos) hn₀
+  . ring_nf
+    exact Nat.ofNat_le_cast.mpr hn₀
+
+lemma imo_1985_p6_unique_top_1
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b)
+  (hd₀ : ∀ (nd : ↑sd), nd.1 + 1 ∈ sd) :
+  ∀ (nd : ↑sd), fd a b nd * (2 - 1 / ↑↑nd) ≤ fd a b ⟨↑nd + 1, hd₀ nd⟩ := by
+  intro nd
+  have hnd₀: 0 < nd.1 := by
+    have g₀: 2 ≤ nd.1 := by
+      refine Set.mem_Ici.mp ?_
+      rw [← hsd]
+      exact nd.2
+    exact Nat.zero_lt_of_lt g₀
+  rw [hfd₁, hfd₁, h₁ nd.1 _ hnd₀, h₁ nd.1 _ hnd₀]
+  have hnd₁: f (↑nd) b * (f (↑nd) b + 1 / ↑↑nd) - f (↑nd) a * (f (↑nd) a + 1 / ↑↑nd) =
+    (f (↑nd) b - f (↑nd) a) * (f (↑nd) b + f (↑nd) a + 1 / nd.1) := by
+    ring_nf
+  rw [hnd₁]
+  refine (mul_le_mul_left ?_).mpr ?_
+  . rw [← hfd₁]
+    exact hd₁ nd a b ha₀
+  . refine le_sub_iff_add_le.mp ?_
+    rw [sub_neg_eq_add]
+    have hnd₂: 1 - 1 / nd.1 < f (↑nd) b := by
+      exact h₇ nd.1 b hnd₀ (ha₁ nd).2
+    have hnd₃: 1 - 1 / nd.1 < f (↑nd) a := by
+      exact h₇ nd.1 a hnd₀ (ha₁ nd).1
+    linarith
+
+lemma imo_1985_p6_unique_top_2
+  (f : ℕ → NNReal → ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b)
+  (nd : ↑sd)
+  (hnd₀ : 0 < nd.1):
+   (f (↑nd) b - f (↑nd) a) * (2 - 1 / ↑↑nd) ≤ f (↑nd) b * (f (↑nd) b + 1 / ↑↑nd) - f (↑nd) a * (f (↑nd) a + 1 / ↑↑nd) := by
+  have hnd₁: f (↑nd) b * (f (↑nd) b + 1 / ↑↑nd) - f (↑nd) a * (f (↑nd) a + 1 / ↑↑nd) =
+      (f (↑nd) b - f (↑nd) a) * (f (↑nd) b + f (↑nd) a + 1 / nd.1) := by
+    ring_nf
+  rw [hnd₁]
+  refine (mul_le_mul_left ?_).mpr ?_
+  . rw [← hfd₁]
+    exact hd₁ nd a b ha₀
+  . refine le_sub_iff_add_le.mp ?_
+    rw [sub_neg_eq_add]
+    have hnd₂: 1 - 1 / nd.1 < f (↑nd) b := by
+      exact h₇ nd.1 b hnd₀ (ha₁ nd).2
+    have hnd₃: 1 - 1 / nd.1 < f (↑nd) a := by
+      exact h₇ nd.1 a hnd₀ (ha₁ nd).1
+    linarith
+
+
+lemma imo_1985_p6_unique_top_3
+  (f : ℕ → NNReal → ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (ha₁ : ∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b)
+  (nd : ↑sd)
+  (hnd₀ : 0 < nd.1)
+  (hnd₁ : f (↑nd) b * (f (↑nd) b + 1 / ↑↑nd) - f (↑nd) a * (f (↑nd) a + 1 / ↑↑nd) =
+    ((f (↑nd) b - f (↑nd) a) * (f (↑nd) b + f (↑nd) a + 1 / ↑↑nd))) :
+   (f (↑nd) b - f (↑nd) a) * (2 - 1 / ↑↑nd) ≤ f (↑nd) b * (f (↑nd) b + 1 / ↑↑nd) - f (↑nd) a * (f (↑nd) a + 1 / ↑↑nd) := by
+  rw [hnd₁]
+  refine (mul_le_mul_left ?_).mpr ?_
+  . rw [← hfd₁]
+    exact hd₁ nd a b ha₀
+  . refine le_sub_iff_add_le.mp ?_
+    rw [sub_neg_eq_add]
+    have hnd₂: 1 - 1 / nd.1 < f (↑nd) b := by
+      exact h₇ nd.1 b hnd₀ (ha₁ nd).2
+    have hnd₃: 1 - 1 / nd.1 < f (↑nd) a := by
+      exact h₇ nd.1 a hnd₀ (ha₁ nd).1
+    linarith
+
+lemma imo_1985_p6_unique_top_4
+  (f : ℕ → NNReal → ℝ)
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (sd : Set ℕ)
+  (a b : NNReal)
+  (ha₁ : ∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b)
+  (nd : ↑sd)
+  (hnd₀ : 0 < nd.1):
+   2 - 1 / ↑↑nd ≤ f (↑nd) b + f (↑nd) a + 1 / ↑↑nd := by
+  refine le_sub_iff_add_le.mp ?_
+  rw [sub_neg_eq_add]
+  have hnd₂: 1 - 1 / nd.1 < f (↑nd) b := by
+    exact h₇ nd.1 b hnd₀ (ha₁ nd).2
+  have hnd₃: 1 - 1 / nd.1 < f (↑nd) a := by
+    exact h₇ nd.1 a hnd₀ (ha₁ nd).1
+  linarith
+
+lemma imo_1985_p6_unique_top_5
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hd₀ : ∀ (nd : ↑sd), nd.1 + 1 ∈ sd)
+  (hd₂ : ∀ (nd : ↑sd), fd a b nd * (2 - 1 / ↑↑nd) ≤ fd a b ⟨↑nd + 1, hd₀ nd⟩)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩) :
+  Filter.Tendsto (fd a b) Filter.atTop Filter.atTop := by
+  have hd₃: ∀ nd, fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd := by
+    intro nd
+    exact imo_1985_p6_unique_top_ind f sd hsd fd hfd₁ hd₁ a b ha₀ hd₀ hd₂ hi i hi₁ nd
+  have hsd₁: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    exact Set.nonempty_of_mem (hd₀ i)
+  refine Filter.tendsto_atTop_atTop.mpr ?_
+  intro z
+  by_cases hz₀: z ≤ fd a b i
+  . use i
+    intros j _
+    refine le_trans hz₀ ?_
+    refine le_trans ?_ (hd₃ j)
+    refine le_mul_of_one_le_right ?_ ?_
+    . refine le_of_lt ?_
+      exact hd₁ i a b ha₀
+    . refine one_le_pow₀ ?_
+      linarith
+  . push_neg at hz₀
+    have hz₁: 0 < fd a b i := by exact hd₁ i a b ha₀
+    have hz₂: 0 < Real.log (z / fd a b i) := by
+      refine Real.log_pos ?_
+      exact (one_lt_div hz₁).mpr hz₀
+    let j : ℕ := Nat.ceil (2 + Real.log (z / fd a b i) / Real.log (3 / 2))
+    have hj₀: 2 < j := by
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    have hj₁: j ∈ sd := by
+      rw [hsd]
+      exact Set.mem_Ici_of_Ioi hj₀
+    use ⟨j, hj₁⟩
+    intro k hk₀
+    have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+      exact hd₃ k
+    have hk₂: i < k := by
+      refine lt_of_lt_of_le ?_ hk₀
+      refine Subtype.mk_lt_mk.mpr ?_
+      refine Nat.lt_ceil.mpr ?_
+      rw [hi₁]
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . rw [← hi₁]
+        exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    refine le_trans ?_ hk₁
+    refine (div_le_iff₀' ?_).mp ?_
+    . exact hz₁
+    . refine Real.le_pow_of_log_le (by linarith) ?_
+      refine (div_le_iff₀ ?_).mp ?_
+      . refine Real.log_pos ?_
+        linarith
+      . rw [Nat.cast_sub ?_]
+        . rw [Nat.cast_two]
+          refine le_sub_iff_add_le'.mpr ?_
+          exact Nat.le_of_ceil_le hk₀
+        . rw [hi₁] at hk₂
+          exact Nat.le_of_succ_le hk₂
+
+
+lemma imo_1985_p6_unique_top_6
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hd₀ : ∀ (nd : ↑sd), nd.1 + 1 ∈ sd)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd):
+   Filter.Tendsto (fd a b) Filter.atTop Filter.atTop := by
+  have hsd₁: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    exact Set.nonempty_of_mem (hd₀ i)
+  refine Filter.tendsto_atTop_atTop.mpr ?_
+  intro z
+  by_cases hz₀: z ≤ fd a b i
+  . use i
+    intros j _
+    refine le_trans hz₀ ?_
+    refine le_trans ?_ (hd₃ j)
+    refine le_mul_of_one_le_right ?_ ?_
+    . refine le_of_lt ?_
+      exact hd₁ i a b ha₀
+    . refine one_le_pow₀ ?_
+      linarith
+  . push_neg at hz₀
+    have hz₁: 0 < fd a b i := by exact hd₁ i a b ha₀
+    have hz₂: 0 < Real.log (z / fd a b i) := by
+      refine Real.log_pos ?_
+      exact (one_lt_div hz₁).mpr hz₀
+    let j : ℕ := Nat.ceil (2 + Real.log (z / fd a b i) / Real.log (3 / 2))
+    have hj₀: 2 < j := by
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    have hj₁: j ∈ sd := by
+      rw [hsd]
+      exact Set.mem_Ici_of_Ioi hj₀
+    use ⟨j, hj₁⟩
+    intro k hk₀
+    have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+      exact hd₃ k
+    have hk₂: i < k := by
+      refine lt_of_lt_of_le ?_ hk₀
+      refine Subtype.mk_lt_mk.mpr ?_
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      rw [hi₁]
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . rw [← hi₁]
+        exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    refine le_trans ?_ hk₁
+    refine (div_le_iff₀' ?_).mp ?_
+    . exact hz₁
+    . refine Real.le_pow_of_log_le (by linarith) ?_
+      refine (div_le_iff₀ ?_).mp ?_
+      . refine Real.log_pos ?_
+        linarith
+      . rw [Nat.cast_sub ?_]
+        . rw [Nat.cast_two]
+          refine le_sub_iff_add_le'.mpr ?_
+          exact Nat.le_of_ceil_le hk₀
+        . rw [hi₁] at hk₂
+          exact Nat.le_of_succ_le hk₂
+
+
+lemma imo_1985_p6_unique_top_7
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd)
+  (hsd₁ : Nonempty ↑sd):
+   Filter.Tendsto (fd a b) Filter.atTop Filter.atTop := by
+  refine Filter.tendsto_atTop_atTop.mpr ?_
+  intro z
+  by_cases hz₀: z ≤ fd a b i
+  . use i
+    intros j _
+    refine le_trans hz₀ ?_
+    refine le_trans ?_ (hd₃ j)
+    refine le_mul_of_one_le_right ?_ ?_
+    . refine le_of_lt ?_
+      exact hd₁ i a b ha₀
+    . refine one_le_pow₀ ?_
+      linarith
+  . push_neg at hz₀
+    have hz₁: 0 < fd a b i := by exact hd₁ i a b ha₀
+    have hz₂: 0 < Real.log (z / fd a b i) := by
+      refine Real.log_pos ?_
+      exact (one_lt_div hz₁).mpr hz₀
+    let j : ℕ := Nat.ceil (2 + Real.log (z / fd a b i) / Real.log (3 / 2))
+    have hj₀: 2 < j := by
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    have hj₁: j ∈ sd := by
+      rw [hsd]
+      exact Set.mem_Ici_of_Ioi hj₀
+    use ⟨j, hj₁⟩
+    intro k hk₀
+    have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+      exact hd₃ k
+    have hk₂: i < k := by
+      refine lt_of_lt_of_le ?_ hk₀
+      refine Subtype.mk_lt_mk.mpr ?_
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      rw [hi₁]
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . rw [← hi₁]
+        exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    refine le_trans ?_ hk₁
+    refine (div_le_iff₀' ?_).mp ?_
+    . exact hz₁
+    . refine Real.le_pow_of_log_le (by linarith) ?_
+      refine (div_le_iff₀ ?_).mp ?_
+      . refine Real.log_pos ?_
+        linarith
+      . rw [Nat.cast_sub ?_]
+        . rw [Nat.cast_two]
+          refine le_sub_iff_add_le'.mpr ?_
+          exact Nat.le_of_ceil_le hk₀
+        . rw [hi₁] at hk₂
+          exact Nat.le_of_succ_le hk₂
+
+lemma imo_1985_p6_unique_top_8
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd)
+  (z : ℝ):
+   ∃ i, ∀ (a_1 : ↑sd), i ≤ a_1 → z ≤ fd a b a_1 := by
+  by_cases hz₀: z ≤ fd a b i
+  . use i
+    intros j _
+    refine le_trans hz₀ ?_
+    refine le_trans ?_ (hd₃ j)
+    refine le_mul_of_one_le_right ?_ ?_
+    . refine le_of_lt ?_
+      exact hd₁ i a b ha₀
+    . refine one_le_pow₀ ?_
+      linarith
+  . push_neg at hz₀
+    have hz₁: 0 < fd a b i := by exact hd₁ i a b ha₀
+    have hz₂: 0 < Real.log (z / fd a b i) := by
+      refine Real.log_pos ?_
+      exact (one_lt_div hz₁).mpr hz₀
+    let j : ℕ := Nat.ceil (2 + Real.log (z / fd a b i) / Real.log (3 / 2))
+    have hj₀: 2 < j := by
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    have hj₁: j ∈ sd := by
+      rw [hsd]
+      exact Set.mem_Ici_of_Ioi hj₀
+    use ⟨j, hj₁⟩
+    intro k hk₀
+    have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+      exact hd₃ k
+    have hk₂: i < k := by
+      refine lt_of_lt_of_le ?_ hk₀
+      refine Subtype.mk_lt_mk.mpr ?_
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      rw [hi₁]
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . rw [← hi₁]
+        exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    refine le_trans ?_ hk₁
+    refine (div_le_iff₀' ?_).mp ?_
+    . exact hz₁
+    . refine Real.le_pow_of_log_le (by linarith) ?_
+      refine (div_le_iff₀ ?_).mp ?_
+      . refine Real.log_pos ?_
+        linarith
+      . rw [Nat.cast_sub ?_]
+        . rw [Nat.cast_two]
+          refine le_sub_iff_add_le'.mpr ?_
+          exact Nat.le_of_ceil_le hk₀
+        . rw [hi₁] at hk₂
+          exact Nat.le_of_succ_le hk₂
+
+
+
+lemma imo_1985_p6_unique_top_9
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (i : ↑sd)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd)
+  (z : ℝ)
+  (hz₀ : z ≤ fd a b i):
+   ∃ i, ∀ (a_1 : ↑sd), i ≤ a_1 → z ≤ fd a b a_1 := by
+  use i
+  intros j _
+  refine le_trans hz₀ ?_
+  refine le_trans ?_ (hd₃ j)
+  refine le_mul_of_one_le_right ?_ ?_
+  . refine le_of_lt ?_
+    exact hd₁ i a b ha₀
+  . refine one_le_pow₀ ?_
+    linarith
+
+lemma imo_1985_p6_unique_top_10
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (i : ↑sd)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd)
+  (z : ℝ)
+  (hz₀ : z ≤ fd a b i):
+   ∀ (a_1 : ↑sd), i ≤ a_1 → z ≤ fd a b a_1 := by
+  intros j _
+  refine le_trans hz₀ ?_
+  refine le_trans ?_ (hd₃ j)
+  refine le_mul_of_one_le_right ?_ ?_
+  . refine le_of_lt ?_
+    exact hd₁ i a b ha₀
+  . refine one_le_pow₀ ?_
+    linarith
+
+lemma imo_1985_p6_unique_top_11
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (i : ↑sd)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd)
+  (j : ↑sd) :
+   fd a b i ≤ fd a b j := by
+  refine le_trans ?_ (hd₃ j)
+  refine le_mul_of_one_le_right ?_ ?_
+  . refine le_of_lt ?_
+    exact hd₁ i a b ha₀
+  . refine one_le_pow₀ ?_
+    linarith
+
+lemma imo_1985_p6_unique_top_12
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd)
+  (z : ℝ)
+  (hz₀ : fd a b i < z):
+   ∃ i, ∀ (a_1 : ↑sd), i ≤ a_1 → z ≤ fd a b a_1 := by
+  have hz₁: 0 < fd a b i := by exact hd₁ i a b ha₀
+  have hz₂: 0 < Real.log (z / fd a b i) := by
+    refine Real.log_pos ?_
+    exact (one_lt_div hz₁).mpr hz₀
+  let j : ℕ := Nat.ceil (2 + Real.log (z / fd a b i) / Real.log (3 / 2))
+  have hj₀: 2 < j := by
+    refine Nat.lt_ceil.mpr ?_
+    norm_cast
+    refine lt_add_of_pos_right 2 ?_
+    refine div_pos ?_ ?_
+    . exact hz₂
+    . refine Real.log_pos ?_
+      linarith
+  have hj₁: j ∈ sd := by
+    rw [hsd]
+    exact Set.mem_Ici_of_Ioi hj₀
+  use ⟨j, hj₁⟩
+  intro k hk₀
+  have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+    exact hd₃ k
+  have hk₂: i < k := by
+    refine lt_of_lt_of_le ?_ hk₀
+    refine Subtype.mk_lt_mk.mpr ?_
+    refine Nat.lt_ceil.mpr ?_
+    norm_cast
+    rw [hi₁]
+    refine lt_add_of_pos_right 2 ?_
+    refine div_pos ?_ ?_
+    . rw [← hi₁]
+      exact hz₂
+    . refine Real.log_pos ?_
+      linarith
+  refine le_trans ?_ hk₁
+  refine (div_le_iff₀' ?_).mp ?_
+  . exact hz₁
+  . refine Real.le_pow_of_log_le (by linarith) ?_
+    refine (div_le_iff₀ ?_).mp ?_
+    . refine Real.log_pos ?_
+      linarith
+    . rw [Nat.cast_sub ?_]
+      . rw [Nat.cast_two]
+        refine le_sub_iff_add_le'.mpr ?_
+        exact Nat.le_of_ceil_le hk₀
+      . rw [hi₁] at hk₂
+        exact Nat.le_of_succ_le hk₂
+
+lemma imo_1985_p6_unique_top_13
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (i : ↑sd)
+  (z : ℝ)
+  (hz₂ : 0 < Real.log (z / fd a b i))
+  (j : ℕ)
+  (hj : j = ⌈2 + Real.log (z / fd a b i) / Real.log (3 / 2)⌉₊):
+  2 < j := by
+  rw [hj]
+  refine Nat.lt_ceil.mpr ?_
+  norm_cast
+  refine lt_add_of_pos_right 2 ?_
+  refine div_pos ?_ ?_
+  . exact hz₂
+  . refine Real.log_pos ?_
+    linarith
+
+lemma imo_1985_p6_unique_top_14
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd)
+  (z : ℝ)
+  (hz₁ : 0 < fd a b i)
+  (hz₂ : 0 < Real.log (z / fd a b i))
+  (j : ℕ)
+  (hj : j = ⌈2 + Real.log (z / fd a b i) / Real.log (3 / 2)⌉₊)
+  (hj₀ : 2 < j):
+   ∃ i, ∀ (a_1 : ↑sd), i ≤ a_1 → z ≤ fd a b a_1 := by
+  have hj₁: j ∈ sd := by
+    rw [hsd]
+    exact Set.mem_Ici_of_Ioi hj₀
+  use ⟨j, hj₁⟩
+  intro k hk₀
+  have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+    exact hd₃ k
+  have hk₂: i < k := by
+    refine lt_of_lt_of_le ?_ hk₀
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hj, hi₁]
+    refine Nat.lt_ceil.mpr ?_
+    refine lt_add_of_pos_right 2 ?_
+    refine div_pos ?_ ?_
+    . rw [← hi₁]
+      exact hz₂
+    . refine Real.log_pos ?_
+      linarith
+  refine le_trans ?_ hk₁
+  refine (div_le_iff₀' ?_).mp ?_
+  . exact hz₁
+  . refine Real.le_pow_of_log_le (by linarith) ?_
+    refine (div_le_iff₀ ?_).mp ?_
+    . refine Real.log_pos ?_
+      linarith
+    . rw [Nat.cast_sub ?_]
+      . rw [Nat.cast_two]
+        refine le_sub_iff_add_le'.mpr ?_
+        refine Nat.le_of_ceil_le ?_
+        exact le_of_eq_of_le (id (Eq.symm hj)) hk₀
+      . rw [hi₁] at hk₂
+        exact Nat.le_of_succ_le hk₂
+
+lemma imo_1985_p6_unique_top_15
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd)
+  (z : ℝ)
+  (hz₁ : 0 < fd a b i)
+  (hz₂ : 0 < Real.log (z / fd a b i))
+  (j : ℕ)
+  (hj : j = ⌈2 + Real.log (z / fd a b i) / Real.log (3 / 2)⌉₊)
+  (hj₁ : j ∈ sd):
+   ∃ i, ∀ (a_1 : ↑sd), i ≤ a_1 → z ≤ fd a b a_1 := by
+  use ⟨j, hj₁⟩
+  intro k hk₀
+  have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+    exact hd₃ k
+  have hk₂: i < k := by
+    refine lt_of_lt_of_le ?_ hk₀
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hj, hi₁]
+    refine Nat.lt_ceil.mpr ?_
+    norm_cast
+    refine lt_add_of_pos_right 2 ?_
+    refine div_pos ?_ ?_
+    . rw [← hi₁]
+      exact hz₂
+    . refine Real.log_pos ?_
+      linarith
+  refine le_trans ?_ hk₁
+  refine (div_le_iff₀' ?_).mp ?_
+  . exact hz₁
+  . refine Real.le_pow_of_log_le (by linarith) ?_
+    refine (div_le_iff₀ ?_).mp ?_
+    . refine Real.log_pos ?_
+      linarith
+    . rw [Nat.cast_sub ?_]
+      . rw [Nat.cast_two]
+        refine le_sub_iff_add_le'.mpr ?_
+        refine Nat.le_of_ceil_le ?_
+        exact le_of_eq_of_le (id (Eq.symm hj)) hk₀
+      . rw [hi₁] at hk₂
+        exact Nat.le_of_succ_le hk₂
+
+lemma imo_1985_p6_unique_top_16
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (hd₃ : ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd)
+  (z : ℝ)
+  (hz₁ : 0 < fd a b i)
+  (hz₂ : 0 < Real.log (z / fd a b i))
+  (j : ℕ)
+  (hj : j = ⌈2 + Real.log (z / fd a b i) / Real.log (3 / 2)⌉₊)
+  (hj₁ : j ∈ sd):
+   ∀ (a_1 : ↑sd), ⟨j, hj₁⟩ ≤ a_1 → z ≤ fd a b a_1 := by
+  intro k hk₀
+  have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+    exact hd₃ k
+  have hk₂: i < k := by
+    refine lt_of_lt_of_le ?_ hk₀
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hj, hi₁]
+    refine Nat.lt_ceil.mpr ?_
+    norm_cast
+    refine lt_add_of_pos_right 2 ?_
+    refine div_pos ?_ ?_
+    . rw [← hi₁]
+      exact hz₂
+    . refine Real.log_pos ?_
+      linarith
+  refine le_trans ?_ hk₁
+  refine (div_le_iff₀' ?_).mp ?_
+  . exact hz₁
+  . refine Real.le_pow_of_log_le (by linarith) ?_
+    refine (div_le_iff₀ ?_).mp ?_
+    . refine Real.log_pos ?_
+      linarith
+    . rw [Nat.cast_sub ?_]
+      . rw [Nat.cast_two]
+        refine le_sub_iff_add_le'.mpr ?_
+        refine Nat.le_of_ceil_le ?_
+        exact le_of_eq_of_le (id (Eq.symm hj)) hk₀
+      . rw [hi₁] at hk₂
+        exact Nat.le_of_succ_le hk₂
+
+
+lemma imo_1985_p6_unique_top_17
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (hi : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi⟩)
+  (z : ℝ)
+  (hz₂ : 0 < Real.log (z / fd a b i))
+  (j : ℕ)
+  (hj : j = ⌈2 + Real.log (z / fd a b i) / Real.log (3 / 2)⌉₊)
+  (hj₁ : j ∈ sd)
+  (k : ↑sd)
+  (hk₀ : ⟨j, hj₁⟩ ≤ k):
+   i < k := by
+  refine lt_of_lt_of_le ?_ hk₀
+  refine Subtype.mk_lt_mk.mpr ?_
+  rw [hj, hi₁]
+  refine Nat.lt_ceil.mpr ?_
+  norm_cast
+  refine lt_add_of_pos_right 2 ?_
+  refine div_pos ?_ ?_
+  . rw [← hi₁]
+    exact hz₂
+  . refine Real.log_pos ?_
+    linarith
+
+lemma imo_1985_p6_unique_top_18
+  (sd : Set ℕ)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (a b : NNReal)
+  (i : ↑sd)
+  (z : ℝ)
+  (hz₂ : 0 < Real.log (z / fd a b i)):
+   2 < 2 + Real.log (z / fd a b i) / Real.log (3 / 2) := by
+  refine lt_add_of_pos_right 2 ?_
+  refine div_pos ?_ ?_
+  . exact hz₂
+  . refine Real.log_pos ?_
+    linarith
+
+
+lemma imo_1985_p6_bonus_5_6
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (g₁ : ((1:ℝ) - (↑↑n)⁻¹) ⊔ 0 = 1 - (↑↑n)⁻¹):
+  ((1:ℝ) - (1 - (↑↑n)⁻¹) ⊔ 0).toNNReal = (↑↑n)⁻¹ := by
+  rw [g₁, ← sub_add, sub_self, zero_add]
+  rw [Real.toNNReal_inv]
+  refine inv_inj.mpr ?_
+  exact NNReal.toNNReal_coe_nat n
+
+
+lemma imo_1985_p6_bonus_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hsn₀ : sn = Set.Ici 1)
+  (hsn₁ : ∀ (n : ↑sn), 0 < n.1)
+  (hfb₀ : fb = fun (n : ↑sn) => fi (↑n) (1 - 1 / ↑↑n))
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x) :
+  1 / 2 ∉ sb := by
+  have g₀: ∀ (n:↑sn), fb n ≠ (1 / 2:NNReal) := by
+    intro n
+    have hfb₄: ∀ n, fb n = fi (n.1) (1 - 1 / ↑↑n) := by
+      rw [hfb₀]
+      simp
+    rw [hfb₄]
+    by_contra! hn₀
+    apply (hf₇ n.1 _ _ (hsn₁ n)).mpr at hn₀
+    contrapose! hn₀
+    clear hn₀
+    refine ne_of_gt ?_
+    rw [hf₂ n.1 _ (hsn₁ n)]
+    induction' n with n hn₀
+    refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+    simp
+    have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+      exact Nat.cast_inv_le_one n
+    rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+    norm_cast
+    rw [hsn₀] at hn₀
+    have hn₁: 1 ≤ n := by exact hn₀
+    have g₁: f 2 2⁻¹ = 3 / 4 := by
+      rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+      rw [NNReal.coe_ofNat]
+      norm_cast
+      ring_nf
+    by_cases hn₂: 4 ≤ n
+    . have hn₃: 1 < f n 2⁻¹ := by
+        refine Nat.le_induction ?_ ?_ n hn₂
+        . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+          ring_nf
+          linarith
+        . intros m hm₀ hm₁
+          refine lt_trans hm₁ ?_
+          refine h₈ m _ (by linarith) ?_ ?_
+          . refine inv_pos.mpr ?_
+            exact zero_lt_two
+          . refine lt_trans ?_ hm₁
+            refine sub_lt_self 1 ?_
+            refine one_div_pos.mpr ?_
+            refine Nat.cast_pos.mpr ?_
+            exact Nat.zero_lt_of_lt hm₀
+      have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+        refine sub_lt_self 1 ?_
+        refine inv_pos.mpr ?_
+        exact Nat.cast_pos'.mpr hn₀
+      exact gt_trans hn₃ hn₄
+    . interval_cases n
+      . rw [h₀]
+        norm_cast
+        rw [inv_one, sub_self 1]
+        refine inv_pos.mpr ?_
+        exact Nat.ofNat_pos'
+      . rw [g₁]
+        ring_nf
+        linarith
+      . rw [h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+  rw [hsb₀]
+  contrapose! g₀
+  exact g₀
+
+
+lemma imo_1985_p6_bonus_6_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hsn₀ : sn = Set.Ici 1)
+  (hsn₁ : ∀ (n : ↑sn), 0 < n.1)
+  (hfb₀ : fb = fun (n:↑sn) => fi (↑n) (1 - 1 / ↑↑n))
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x):
+   ∀ (n : ↑sn), fb n ≠ 1 / 2 := by
+  intro n
+  have hfb₄: ∀ n, fb n = fi (n.1) (1 - 1 / ↑↑n) := by
+    rw [hfb₀]
+    simp
+  rw [hfb₄]
+  by_contra! hn₀
+  apply (hf₇ n.1 _ _ (hsn₁ n)).mpr at hn₀
+  contrapose! hn₀
+  clear hn₀
+  refine ne_of_gt ?_
+  rw [hf₂ n.1 _ (hsn₁ n)]
+  induction' n with n hn₀
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  simp
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  norm_cast
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+lemma imo_1985_p6_bonus_6_2
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (g₀ : ∀ (n : ↑sn), fb n ≠ 1 / 2):
+   1 / 2 ∉ sb := by
+  rw [hsb₀]
+  contrapose! g₀
+  exact g₀
+
+
+lemma imo_1985_p6_bonus_6_3
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (hsn₁ : ∀ (n : ↑sn), 0 < n.1)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ↑sn):
+   fi (↑n) (1 - 1 / ↑↑n) ≠ 1 / 2 := by
+  by_contra! hn₀
+  apply (hf₇ n.1 _ _ (hsn₁ n)).mpr at hn₀
+  contrapose! hn₀
+  clear hn₀
+  refine ne_of_gt ?_
+  rw [hf₂ n.1 _ (hsn₁ n)]
+  induction' n with n hn₀
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  simp
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  norm_cast
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+
+lemma imo_1985_p6_bonus_6_4
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (hsn₁ : ∀ (n : ↑sn), 0 < n.1)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ↑sn)
+  (hn₀ : fi (↑n) (1 - 1 / ↑↑n) = 1 / 2):
+   False := by
+  apply (hf₇ n.1 _ _ (hsn₁ n)).mpr at hn₀
+  contrapose! hn₀
+  clear hn₀
+  refine ne_of_gt ?_
+  rw [hf₂ n.1 _ (hsn₁ n)]
+  induction' n with n hn₀
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  simp
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  norm_cast
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+lemma imo_1985_p6_bonus_6_5
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (hsn₁ : ∀ (n : ↑sn), 0 < n.1)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ↑sn):
+   f₀ (↑n) (1 / 2) ≠ 1 - 1 / ↑↑n := by
+  refine ne_of_gt ?_
+  rw [hf₂ n.1 _ (hsn₁ n)]
+  induction' n with n hn₀
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  simp
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  norm_cast
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+
+lemma imo_1985_p6_bonus_6_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (hsn₁ : ∀ (n : ↑sn), 0 < n.1)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ↑sn):
+   1 - 1 / ↑↑n < f₀ (↑n) (1 / 2) := by
+  rw [hf₂ n.1 _ (hsn₁ n)]
+  induction' n with n hn₀
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  simp
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  norm_cast
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+lemma imo_1985_p6_bonus_6_7
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ↑sn):
+   1 - 1 / ↑↑n < (f (↑n) (1 / 2)).toNNReal := by
+  induction' n with n hn₀
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  simp
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  norm_cast
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+lemma imo_1985_p6_bonus_6_8
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ℕ)
+  (hn₀ : n ∈ sn):
+  (1:NNReal) - 1 / ↑n < (f n (1 / 2)).toNNReal := by
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  simp
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  norm_cast
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+lemma imo_1985_p6_bonus_6_9
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ℕ)
+  (hn₀ : n ∈ sn):
+   ↑((1:NNReal) - 1 / n) < f n (1 / 2) := by
+  simp
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  norm_cast
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+lemma imo_1985_p6_bonus_6_10
+  (f : ℕ → NNReal → ℝ)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hsn₁ : ∀ (n : ↑sn), 0 < n.1)
+  (n : ↑sn)
+  (hfb₄ : ∀ (n : ↑sn), fb n = fi (↑n) (1 - 1 / ↑↑n))
+  (hfb₅: ↑((1:NNReal) - 1 / n) < f n (1 / 2)):
+   fb n ≠ 1 / 2 := by
+  rw [hfb₄]
+  by_contra! hn₀
+  apply (hf₇ n.1 _ _ (hsn₁ n)).mpr at hn₀
+  contrapose! hn₀
+  clear hn₀
+  refine ne_of_gt ?_
+  rw [hf₂ n.1 _ (hsn₁ n)]
+  induction' n with n hn₀
+  exact Real.lt_toNNReal_iff_coe_lt.mpr hfb₅
+
+
+lemma imo_1985_p6_bonus_6_11
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ℕ)
+  (hn₀ : n ∈ sn):
+  ↑((1:NNReal) - (↑n)⁻¹) < f n 2⁻¹ := by
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  norm_cast
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+lemma imo_1985_p6_bonus_6_12
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ℕ)
+  (hn₀ : n ∈ sn)
+  (g₀ : (↑n)⁻¹ ≤ (1:NNReal)):
+   1 - (↑n)⁻¹ < f n 2⁻¹ := by
+  rw [hsn₀] at hn₀
+  have hn₁: 1 ≤ n := by exact hn₀
+  have g₁: f 2 2⁻¹ = 3 / 4 := by
+    rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+    rw [NNReal.coe_ofNat]
+    norm_cast
+    ring_nf
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+lemma imo_1985_p6_bonus_6_13
+  (f : ℕ → NNReal → ℝ)
+  (n : ℕ)
+  (hn₁ : 1 - (↑n)⁻¹ < f n 2⁻¹):
+   ↑((1:NNReal) - (↑n)⁻¹) < f n 2⁻¹ := by
+  have g₀: (↑n)⁻¹ ≤ (1:NNReal) := by
+    exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv, NNReal.coe_natCast]
+  exact hn₁
+
+
+lemma imo_1985_p6_bonus_6_14
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n)):
+   f 2 2⁻¹ = 3 / 4 := by
+  rw [h₁ 1 _ (by linarith), h₀, inv_eq_one_div 2, NNReal.coe_div 1 2]
+  rw [NNReal.coe_ofNat]
+  norm_cast
+  ring_nf
+
+
+lemma imo_1985_p6_bonus_6_15
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ℕ)
+  (hn₀ : n ∈ Set.Ici 1)
+  (g₀ : (↑n)⁻¹ ≤ (1:NNReal))
+  (hn₁ : 1 ≤ n)
+  (g₁ : f 2 2⁻¹ = 3 / 4):
+   1 - (↑n)⁻¹ < f n 2⁻¹ := by
+  by_cases hn₂: 4 ≤ n
+  . have hn₃: 1 < f n 2⁻¹ := by
+      refine Nat.le_induction ?_ ?_ n hn₂
+      . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+        ring_nf
+        linarith
+      . intros m hm₀ hm₁
+        refine lt_trans hm₁ ?_
+        refine h₈ m _ (by linarith) ?_ ?_
+        . refine inv_pos.mpr ?_
+          exact zero_lt_two
+        . refine lt_trans ?_ hm₁
+          refine sub_lt_self 1 ?_
+          refine one_div_pos.mpr ?_
+          refine Nat.cast_pos.mpr ?_
+          exact Nat.zero_lt_of_lt hm₀
+    have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+      refine sub_lt_self 1 ?_
+      refine inv_pos.mpr ?_
+      exact Nat.cast_pos'.mpr hn₀
+    exact gt_trans hn₃ hn₄
+  . interval_cases n
+    . rw [h₀]
+      norm_cast
+      rw [inv_one, sub_self 1]
+      refine inv_pos.mpr ?_
+      exact Nat.ofNat_pos'
+    . rw [g₁]
+      ring_nf
+      linarith
+    . rw [h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+
+
+lemma imo_1985_p6_bonus_6_16
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ℕ)
+  (hn₀ : n ∈ Set.Ici 1)
+  (g₁ : f 2 2⁻¹ = 3 / 4)
+  (hn₂ : 4 ≤ n):
+   1 - (↑n)⁻¹ < f n 2⁻¹ := by
+  have hn₃: 1 < f n 2⁻¹ := by
+    refine Nat.le_induction ?_ ?_ n hn₂
+    . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+      ring_nf
+      linarith
+    . intros m hm₀ hm₁
+      refine lt_trans hm₁ ?_
+      refine h₈ m _ (by linarith) ?_ ?_
+      . refine inv_pos.mpr ?_
+        exact zero_lt_two
+      . refine lt_trans ?_ hm₁
+        refine sub_lt_self 1 ?_
+        refine one_div_pos.mpr ?_
+        refine Nat.cast_pos.mpr ?_
+        exact Nat.zero_lt_of_lt hm₀
+  have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+    refine sub_lt_self 1 ?_
+    refine inv_pos.mpr ?_
+    exact Nat.cast_pos'.mpr hn₀
+  exact gt_trans hn₃ hn₄
+
+lemma imo_1985_p6_bonus_6_17
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (n : ℕ)
+  (g₁ : f 2 2⁻¹ = 3 / 4)
+  (hn₂ : 4 ≤ n):
+   1 < f n 2⁻¹ := by
+  refine Nat.le_induction ?_ ?_ n hn₂
+  . rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+    ring_nf
+    linarith
+  . intros m hm₀ hm₁
+    refine lt_trans hm₁ ?_
+    refine h₈ m _ (by linarith) ?_ ?_
+    . refine inv_pos.mpr ?_
+      exact zero_lt_two
+    . refine lt_trans ?_ hm₁
+      refine sub_lt_self 1 ?_
+      refine one_div_pos.mpr ?_
+      refine Nat.cast_pos.mpr ?_
+      exact Nat.zero_lt_of_lt hm₀
+
+
+lemma imo_1985_p6_bonus_6_18
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (g₁ : f 2 2⁻¹ = 3 / 4):
+   1 < f 4 2⁻¹ := by
+  rw [h₁ 3 _ (by linarith), h₁ 2 _ (by linarith), g₁]
+  ring_nf
+  linarith
+
+
+lemma imo_1985_p6_bonus_6_19
+  (f : ℕ → NNReal → ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x):
+  ∀ (n : ℕ), 4 ≤ n → 1 < f n 2⁻¹ → 1 < f (n + 1) 2⁻¹ := by
+  intros m hm₀ hm₁
+  refine lt_trans hm₁ ?_
+  refine h₈ m _ (by linarith) ?_ ?_
+  . refine inv_pos.mpr ?_
+    exact zero_lt_two
+  . refine lt_trans ?_ hm₁
+    refine sub_lt_self 1 ?_
+    refine one_div_pos.mpr ?_
+    refine Nat.cast_pos.mpr ?_
+    exact Nat.zero_lt_of_lt hm₀
+
+
+lemma imo_1985_p6_bonus_6_20
+  (f : ℕ → NNReal → ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (m : ℕ)
+  (hm₀ : 4 ≤ m)
+  (hm₁ : 1 < f m 2⁻¹):
+   f m 2⁻¹ < f (m + 1) 2⁻¹ := by
+  refine h₈ m _ (by linarith) ?_ ?_
+  . refine inv_pos.mpr ?_
+    exact zero_lt_two
+  . refine lt_trans ?_ hm₁
+    refine sub_lt_self 1 ?_
+    refine one_div_pos.mpr ?_
+    refine Nat.cast_pos.mpr ?_
+    exact Nat.zero_lt_of_lt hm₀
+
+lemma imo_1985_p6_bonus_6_21
+  (f : ℕ → NNReal → ℝ)
+  (m : ℕ)
+  (hm₀ : 4 ≤ m)
+  (hm₁ : 1 < f m 2⁻¹):
+   1 - 1 / ↑m < f m 2⁻¹ := by
+  refine lt_trans ?_ hm₁
+  refine sub_lt_self 1 ?_
+  refine one_div_pos.mpr ?_
+  refine Nat.cast_pos.mpr ?_
+  exact Nat.zero_lt_of_lt hm₀
+
+lemma imo_1985_p6_bonus_6_22
+  (f : ℕ → NNReal → ℝ)
+  (n : ℕ)
+  (hn₀ : n ∈ Set.Ici 1)
+  (hn₃ : 1 < f n 2⁻¹):
+   1 - (↑n)⁻¹ < f n 2⁻¹ := by
+  have hn₄: (1:ℝ) - (↑n)⁻¹ < 1 := by
+    refine sub_lt_self 1 ?_
+    refine inv_pos.mpr ?_
+    exact Nat.cast_pos'.mpr hn₀
+  exact gt_trans hn₃ hn₄
+
+lemma imo_1985_p6_bonus_6_23
+  (n : ℕ)
+  (hn₀ : n ∈ Set.Ici 1):
+  (1:ℝ) - (↑n)⁻¹ < 1 := by
+  refine sub_lt_self 1 ?_
+  refine inv_pos.mpr ?_
+  exact Nat.cast_pos'.mpr hn₀
+
+lemma imo_1985_p6_bonus_6_24
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (n : ℕ)
+  (hn₀ : n ∈ Set.Ici 1)
+  (g₀ : (↑n)⁻¹ ≤ (1:NNReal))
+  (hn₁ : 1 ≤ n)
+  (g₁ : f 2 2⁻¹ = 3 / 4)
+  (hn₂ : ¬4 ≤ n):
+   1 - (↑n)⁻¹ < f n 2⁻¹ := by
+  interval_cases n
+  . rw [h₀]
+    norm_cast
+    rw [inv_one, sub_self 1]
+    refine inv_pos.mpr ?_
+    exact Nat.ofNat_pos'
+  . rw [g₁]
+    ring_nf
+    linarith
+  . rw [h₁ 2 _ (by linarith), g₁]
+    ring_nf
+    linarith
+
+lemma imo_1985_p6_bonus_6_25
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x):
+   1 - (↑1)⁻¹ < f 1 2⁻¹ := by
+  rw [h₀]
+  norm_cast
+  rw [inv_one, sub_self 1]
+  refine inv_pos.mpr ?_
+  exact Nat.ofNat_pos'
+
+lemma imo_1985_p6_bonus_6_26
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (g₁ : f 2 2⁻¹ = 3 / 4):
+   1 - (↑3)⁻¹ < f 3 2⁻¹ := by
+  rw [h₁ 2 _ (by linarith), g₁]
+  ring_nf
+  linarith
+
+
+lemma imo_1985_p6_10_16
+  (sn : Set ℕ)
+  (sb : Set NNReal)
+  (fb : ↑sn → NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (hnb₀ : 2 ∈ sn)
+  (nb : ↑sn := ⟨2, hnb₀⟩) :
+   ∃ x ∈ sb, fr x = ↑(fb nb) := by
+  use fb ↑nb
+  constructor
+  . rw [hsb₀]
+    exact Set.mem_range_self nb
+  . exact congrFun hfr (fb ↑nb)
+
+
+lemma imo_1985_p6_10_17
+  (sn : Set ℕ)
+  (sb : Set NNReal)
+  (fb : ↑sn → NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (hnb₀ : 2 ∈ sn)
+  (nb : ↑sn := ⟨2, hnb₀⟩):
+   fb nb ∈ sb ∧ fr (fb nb) = ↑(fb nb) := by
+  constructor
+  . rw [hsb₀]
+    exact Set.mem_range_self nb
+  . exact congrFun hfr (fb ↑nb)
+
+
+lemma imo_1985_p6_10_18
+  (sbr : Set ℝ)
+  (br : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (g₁ : ∃ x, 0 < x ∧ x ∈ sbr):
+   0 < br := by
+  obtain ⟨x, hx₀, hx₁⟩ := g₁
+  have hx₂: br ∈ upperBounds sbr := by
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  exact gt_of_ge_of_gt (hx₂ hx₁) hx₀
+
+
+lemma imo_1985_p6_10_19
+  (sbr : Set ℝ)
+  (br : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (x : ℝ)
+  (hx₀ : 0 < x)
+  (hx₁ : x ∈ sbr):
+   0 < br := by
+  have hx₂: br ∈ upperBounds sbr := by
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  exact gt_of_ge_of_gt (hx₂ hx₁) hx₀
+
+
+lemma imo_1985_p6_10_20
+  (sbr : Set ℝ)
+  (br : ℝ)
+  (hbr₀ : IsLUB sbr br):
+   br ∈ upperBounds sbr := by
+  refine (isLUB_le_iff hbr₀).mp ?_
+  exact Preorder.le_refl br
+
+
+
+
+lemma imo_1985_p6_11_1
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc):
+   ∀ (nb nc : ↑sn), fb nb < fc nc := by
+  intros nb nc
+  cases' (lt_or_le nb nc) with hn₀ hn₀
+  . refine lt_trans ?_ (hfc₂ nc)
+    exact hfb₃ hn₀
+  cases' lt_or_eq_of_le hn₀ with hn₁ hn₁
+  . refine lt_trans (hfc₂ nb) ?_
+    exact hfc₃ hn₁
+  . rw [hn₁]
+    exact hfc₂ nb
+
+lemma imo_1985_p6_11_2
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (nb nc : ↑sn)
+  (hn₀ : nb < nc):
+   fb nb < fc nc := by
+  refine lt_trans ?_ (hfc₂ nc)
+  exact hfb₃ hn₀
+
+lemma imo_1985_p6_11_3
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfc₃ : StrictAnti fc)
+  (nb nc : ↑sn)
+  (hn₀ : nc ≤ nb):
+   fb nb < fc nc := by
+  cases' lt_or_eq_of_le hn₀ with hn₁ hn₁
+  . refine lt_trans (hfc₂ nb) ?_
+    exact hfc₃ hn₁
+  . rw [hn₁]
+    exact hfc₂ nb
+
+lemma imo_1985_p6_11_4
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (hcr₀ : IsGLB scr cr)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (hfc₄ : ∀ (nb nc : ↑sn), fb nb < fc nc):
+   br ≤ cr := by
+  by_contra! hc₀
+  have hc₁: ∃ x ∈ sbr, cr < x ∧ x ≤ br := by exact IsLUB.exists_between hbr₀ hc₀
+  let ⟨x, hx₀, hx₁, _⟩ := hc₁
+  have hc₂: ∃ y ∈ scr, cr ≤ y ∧ y < x := by exact IsGLB.exists_between hcr₀ hx₁
+  let ⟨y, hy₀, _, hy₂⟩ := hc₂
+  have hc₃: x < y := by
+    have hx₃: x.toNNReal ∈ sb := by
+      rw [hsbr] at hx₀
+      apply (Set.mem_image fr sb x).mp at hx₀
+      obtain ⟨z, hz₀, hz₁⟩ := hx₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    have hy₃: y.toNNReal ∈ sc := by
+      rw [hscr] at hy₀
+      apply (Set.mem_image fr sc y).mp at hy₀
+      obtain ⟨z, hz₀, hz₁⟩ := hy₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    rw [hsb₀] at hx₃
+    rw [hsc₀] at hy₃
+    apply Set.mem_range.mp at hx₃
+    apply Set.mem_range.mp at hy₃
+    let ⟨nx, hnx₀⟩ := hx₃
+    let ⟨ny, hny₀⟩ := hy₃
+    have hy₄: 0 < y := by
+      contrapose! hy₃
+      have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+      intro z
+      rw [hy₅]
+      refine ne_of_gt ?_
+      refine lt_of_le_of_lt ?_ (hfc₂ z)
+      exact hfb₄ z
+    refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+    rw [← hnx₀, ← hny₀]
+    exact hfc₄ nx ny
+  refine (lt_self_iff_false x).mp ?_
+  exact lt_trans hc₃ hy₂
+
+lemma imo_1985_p6_11_5
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (hcr₀ : IsGLB scr cr)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (hfc₄ : ∀ (nb nc : ↑sn), fb nb < fc nc)
+  (hc₁ : ∃ x ∈ sbr, cr < x ∧ x ≤ br):
+   False := by
+  let ⟨x, hx₀, hx₁, _⟩ := hc₁
+  have hc₂: ∃ y ∈ scr, cr ≤ y ∧ y < x := by exact IsGLB.exists_between hcr₀ hx₁
+  let ⟨y, hy₀, _, hy₂⟩ := hc₂
+  have hc₃: x < y := by
+    have hx₃: x.toNNReal ∈ sb := by
+      rw [hsbr] at hx₀
+      apply (Set.mem_image fr sb x).mp at hx₀
+      obtain ⟨z, hz₀, hz₁⟩ := hx₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    have hy₃: y.toNNReal ∈ sc := by
+      rw [hscr] at hy₀
+      apply (Set.mem_image fr sc y).mp at hy₀
+      obtain ⟨z, hz₀, hz₁⟩ := hy₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    rw [hsb₀] at hx₃
+    rw [hsc₀] at hy₃
+    apply Set.mem_range.mp at hx₃
+    apply Set.mem_range.mp at hy₃
+    let ⟨nx, hnx₀⟩ := hx₃
+    let ⟨ny, hny₀⟩ := hy₃
+    have hy₄: 0 < y := by
+      contrapose! hy₃
+      have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+      intro z
+      rw [hy₅]
+      refine ne_of_gt ?_
+      refine lt_of_le_of_lt ?_ (hfc₂ z)
+      exact hfb₄ z
+    refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+    rw [← hnx₀, ← hny₀]
+    exact hfc₄ nx ny
+  refine (lt_self_iff_false x).mp ?_
+  exact lt_trans hc₃ hy₂
+
+
+lemma imo_1985_p6_11_6
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (hfc₄ : ∀ (nb nc : ↑sn), fb nb < fc nc)
+  (x : ℝ)
+  (hx₀ : x ∈ sbr)
+  (y : ℝ)
+  (hy₀ : y ∈ scr)
+  (hy₂ : y < x):
+   False := by
+  have hc₃: x < y := by
+    have hx₃: x.toNNReal ∈ sb := by
+      rw [hsbr] at hx₀
+      apply (Set.mem_image fr sb x).mp at hx₀
+      obtain ⟨z, hz₀, hz₁⟩ := hx₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    have hy₃: y.toNNReal ∈ sc := by
+      rw [hscr] at hy₀
+      apply (Set.mem_image fr sc y).mp at hy₀
+      obtain ⟨z, hz₀, hz₁⟩ := hy₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    rw [hsb₀] at hx₃
+    rw [hsc₀] at hy₃
+    apply Set.mem_range.mp at hx₃
+    apply Set.mem_range.mp at hy₃
+    let ⟨nx, hnx₀⟩ := hx₃
+    let ⟨ny, hny₀⟩ := hy₃
+    have hy₄: 0 < y := by
+      contrapose! hy₃
+      have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+      intro z
+      rw [hy₅]
+      refine ne_of_gt ?_
+      refine lt_of_le_of_lt ?_ (hfc₂ z)
+      exact hfb₄ z
+    refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+    rw [← hnx₀, ← hny₀]
+    exact hfc₄ nx ny
+  refine (lt_self_iff_false x).mp ?_
+  exact lt_trans hc₃ hy₂
+
+lemma imo_1985_p6_11_7
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (hfc₄ : ∀ (nb nc : ↑sn), fb nb < fc nc)
+  (x : ℝ)
+  (hx₀ : x ∈ sbr)
+  (y : ℝ)
+  (hy₀ : y ∈ scr):
+   x < y := by
+  have hx₃: x.toNNReal ∈ sb := by
+    rw [hsbr] at hx₀
+    apply (Set.mem_image fr sb x).mp at hx₀
+    obtain ⟨z, hz₀, hz₁⟩ := hx₀
+    rw [← hz₁, hfr, Real.toNNReal_coe]
+    exact hz₀
+  have hy₃: y.toNNReal ∈ sc := by
+    rw [hscr] at hy₀
+    apply (Set.mem_image fr sc y).mp at hy₀
+    obtain ⟨z, hz₀, hz₁⟩ := hy₀
+    rw [← hz₁, hfr, Real.toNNReal_coe]
+    exact hz₀
+  rw [hsb₀] at hx₃
+  rw [hsc₀] at hy₃
+  apply Set.mem_range.mp at hx₃
+  apply Set.mem_range.mp at hy₃
+  let ⟨nx, hnx₀⟩ := hx₃
+  let ⟨ny, hny₀⟩ := hy₃
+  have hy₄: 0 < y := by
+    contrapose! hy₃
+    have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+    intro z
+    rw [hy₅]
+    refine ne_of_gt ?_
+    refine lt_of_le_of_lt ?_ (hfc₂ z)
+    exact hfb₄ z
+  refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+  rw [← hnx₀, ← hny₀]
+  exact hfc₄ nx ny
+
+lemma imo_1985_p6_11_8
+  (sb : Set NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (x : ℝ)
+  (hx₀ : x ∈ sbr):
+   x.toNNReal ∈ sb := by
+  rw [hsbr] at hx₀
+  apply (Set.mem_image fr sb x).mp at hx₀
+  obtain ⟨z, hz₀, hz₁⟩ := hx₀
+  rw [← hz₁, hfr, Real.toNNReal_coe]
+  exact hz₀
+
+lemma imo_1985_p6_11_9
+  (sb : Set NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (x : ℝ)
+  (hx₀ : ∃ x_1 ∈ sb, fr x_1 = x):
+   x.toNNReal ∈ sb := by
+  obtain ⟨z, hz₀, hz₁⟩ := hx₀
+  rw [← hz₁, hfr, Real.toNNReal_coe]
+  exact hz₀
+
+lemma imo_1985_p6_11_10
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (hfc₄ : ∀ (nb nc : ↑sn), fb nb < fc nc)
+  (x : ℝ)
+  (y : ℝ)
+  (hy₀ : y ∈ scr)
+  (hx₃ : x.toNNReal ∈ sb):
+   x < y := by
+  have hy₃: y.toNNReal ∈ sc := by
+    rw [hscr] at hy₀
+    apply (Set.mem_image fr sc y).mp at hy₀
+    obtain ⟨z, hz₀, hz₁⟩ := hy₀
+    rw [← hz₁, hfr, Real.toNNReal_coe]
+    exact hz₀
+  rw [hsb₀] at hx₃
+  rw [hsc₀] at hy₃
+  apply Set.mem_range.mp at hx₃
+  apply Set.mem_range.mp at hy₃
+  let ⟨nx, hnx₀⟩ := hx₃
+  let ⟨ny, hny₀⟩ := hy₃
+  have hy₄: 0 < y := by
+    contrapose! hy₃
+    have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+    intro z
+    rw [hy₅]
+    refine ne_of_gt ?_
+    refine lt_of_le_of_lt ?_ (hfc₂ z)
+    exact hfb₄ z
+  refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+  rw [← hnx₀, ← hny₀]
+  exact hfc₄ nx ny
+
+lemma imo_1985_p6_11_11
+  (sc : Set NNReal)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (y : ℝ)
+  (hy₀ : y ∈ scr):
+   y.toNNReal ∈ sc := by
+  rw [hscr] at hy₀
+  apply (Set.mem_image fr sc y).mp at hy₀
+  obtain ⟨z, hz₀, hz₁⟩ := hy₀
+  rw [← hz₁, hfr, Real.toNNReal_coe]
+  exact hz₀
+
+
+lemma imo_1985_p6_11_12
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (hfc₄ : ∀ (nb nc : ↑sn), fb nb < fc nc)
+  (x : ℝ)
+  (y : ℝ)
+  (hx₃ : x.toNNReal ∈ sb)
+  (hy₃ : y.toNNReal ∈ sc):
+   x < y := by
+  rw [hsb₀] at hx₃
+  rw [hsc₀] at hy₃
+  apply Set.mem_range.mp at hx₃
+  apply Set.mem_range.mp at hy₃
+  let ⟨nx, hnx₀⟩ := hx₃
+  let ⟨ny, hny₀⟩ := hy₃
+  have hy₄: 0 < y := by
+    contrapose! hy₃
+    have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+    intro z
+    rw [hy₅]
+    refine ne_of_gt ?_
+    refine lt_of_le_of_lt ?_ (hfc₂ z)
+    exact hfb₄ z
+  refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+  rw [← hnx₀, ← hny₀]
+  exact hfc₄ nx ny
+
+lemma imo_1985_p6_11_13
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (hfc₄ : ∀ (nb nc : ↑sn), fb nb < fc nc)
+  (x : ℝ)
+  (y : ℝ)
+  (hx₃ : ∃ y, fb y = x.toNNReal)
+  (hy₃ : ∃ y_1, fc y_1 = y.toNNReal):
+   x < y := by
+  let ⟨nx, hnx₀⟩ := hx₃
+  let ⟨ny, hny₀⟩ := hy₃
+  have hy₄: 0 < y := by
+    contrapose! hy₃
+    have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+    intro z
+    rw [hy₅]
+    refine ne_of_gt ?_
+    refine lt_of_le_of_lt ?_ (hfc₂ z)
+    exact hfb₄ z
+  refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+  rw [← hnx₀, ← hny₀]
+  exact hfc₄ nx ny
+
+lemma imo_1985_p6_11_14
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (y : ℝ)
+  (hy₃ : ∃ y_1, fc y_1 = y.toNNReal):
+   0 < y := by
+  contrapose! hy₃
+  have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+  intro z
+  rw [hy₅]
+  refine ne_of_gt ?_
+  refine lt_of_le_of_lt ?_ (hfc₂ z)
+  exact hfb₄ z
+
+lemma imo_1985_p6_11_15
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (y : ℝ)
+  (hy₃ : y ≤ 0):
+   ∀ (y_1 : ↑sn), fc y_1 ≠ y.toNNReal := by
+  have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+  intro z
+  rw [hy₅]
+  refine ne_of_gt ?_
+  refine lt_of_le_of_lt ?_ (hfc₂ z)
+  exact hfb₄ z
+
+lemma imo_1985_p6_11_16
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n)
+  (y : ℝ)
+  (hy₅ : y.toNNReal = 0)
+  (z : ↑sn):
+   fc z ≠ y.toNNReal := by
+  rw [hy₅]
+  refine ne_of_gt ?_
+  refine lt_of_le_of_lt ?_ (hfc₂ z)
+  exact hfb₄ z
+
+
+lemma imo_1985_p6_11_17
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₄ : ∀ (nb nc : ↑sn), fb nb < fc nc)
+  (x : ℝ)
+  (y : ℝ)
+  (nx : ↑sn)
+  (hnx₀ : fb nx = x.toNNReal)
+  (ny : ↑sn)
+  (hny₀ : fc ny = y.toNNReal)
+  (hy₄ : 0 < y):
+   x < y := by
+  refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+  rw [← hnx₀, ← hny₀]
+  exact hfc₄ nx ny
+
+
+
+lemma imo_1985_p6_exists_1
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (hu₆ : br < cr):
+   ∃ x, ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1 := by
+  apply exists_between at hu₆
+  let ⟨a, ha₀, ha₁⟩ := hu₆
+  have ha₂: 0 < a := by exact gt_trans ha₀ hbr₁
+  have ha₃: 0 < a.toNNReal := by exact Real.toNNReal_pos.mpr ha₂
+  use a.toNNReal
+  intros n hn₀
+  have hn₁: n ∈ sn := by
+    rw [hsn₀]
+    exact hn₀
+  constructor
+  . exact h₂ n a.toNNReal ⟨hn₀, ha₃⟩
+  constructor
+  . refine h₈ n a.toNNReal hn₀ ?_ ?_
+    . exact Real.toNNReal_pos.mpr ha₂
+    . let nn : ↑sn := ⟨n, hn₁⟩
+      have hn₂: f n (fb nn) = 1 - 1 / n := by
+        rw [hf₁ n _ hn₀, hfb₁ nn]
+        refine NNReal.coe_sub ?_
+        refine div_le_self ?_ ?_
+        . exact zero_le_one' NNReal
+        . exact Nat.one_le_cast.mpr hn₀
+      rw [← hn₂]
+      refine hmo₀ n hn₀ ?_
+      refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+      refine lt_of_le_of_lt ?_ ha₀
+      refine hbr₃ _ ?_
+      rw [hsbr]
+      refine (Set.mem_image fr sb _).mpr ?_
+      use (fb nn)
+      rw [hfr, hsb₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self nn
+  . have hn₂: n + 1 ∈ sn := by
+      rw [hsn₀]
+      exact Set.mem_Ici.mpr (by linarith)
+    let nn : ↑sn := ⟨n + 1, hn₂⟩
+    have hn₃: f (n + 1) (fc (nn)) = 1 := by
+      rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+      exact rfl
+    rw [← hn₃]
+    refine hmo₀ (n + 1) (by linarith) ?_
+    refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+    refine lt_of_lt_of_le ha₁ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+
+
+lemma imo_1985_p6_exists_2
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (a : ℝ)
+  (ha₀ : br < a)
+  (ha₁ : a < cr)
+  (ha₂ : 0 < a)
+  (ha₃ : 0 < a.toNNReal):
+   ∃ x, ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1 := by
+  use a.toNNReal
+  intros n hn₀
+  have hn₁: n ∈ sn := by
+    rw [hsn₀]
+    exact hn₀
+  constructor
+  . exact h₂ n a.toNNReal ⟨hn₀, ha₃⟩
+  constructor
+  . refine h₈ n a.toNNReal hn₀ ?_ ?_
+    . exact Real.toNNReal_pos.mpr ha₂
+    . let nn : ↑sn := ⟨n, hn₁⟩
+      have hn₂: f n (fb nn) = 1 - 1 / n := by
+        rw [hf₁ n _ hn₀, hfb₁ nn]
+        refine NNReal.coe_sub ?_
+        refine div_le_self ?_ ?_
+        . exact zero_le_one' NNReal
+        . exact Nat.one_le_cast.mpr hn₀
+      rw [← hn₂]
+      refine hmo₀ n hn₀ ?_
+      refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+      refine lt_of_le_of_lt ?_ ha₀
+      refine hbr₃ _ ?_
+      rw [hsbr]
+      refine (Set.mem_image fr sb _).mpr ?_
+      use (fb nn)
+      rw [hfr, hsb₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self nn
+  . have hn₂: n + 1 ∈ sn := by
+      rw [hsn₀]
+      exact Set.mem_Ici.mpr (by linarith)
+    let nn : ↑sn := ⟨n + 1, hn₂⟩
+    have hn₃: f (n + 1) (fc (nn)) = 1 := by
+      rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+      exact rfl
+    rw [← hn₃]
+    refine hmo₀ (n + 1) (by linarith) ?_
+    refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+    refine lt_of_lt_of_le ha₁ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_3
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (a : ℝ)
+  (ha₀ : br < a)
+  (ha₁ : a < cr)
+  (ha₂ : 0 < a)
+  (ha₃ : 0 < a.toNNReal)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   0 < f n a.toNNReal ∧ f n a.toNNReal < f (n + 1) a.toNNReal ∧ f (n + 1) a.toNNReal < 1 := by
+  have hn₁: n ∈ sn := by
+    rw [hsn₀]
+    exact hn₀
+  constructor
+  . exact h₂ n a.toNNReal ⟨hn₀, ha₃⟩
+  constructor
+  . refine h₈ n a.toNNReal hn₀ ?_ ?_
+    . exact Real.toNNReal_pos.mpr ha₂
+    . let nn : ↑sn := ⟨n, hn₁⟩
+      have hn₂: f n (fb nn) = 1 - 1 / n := by
+        rw [hf₁ n _ hn₀, hfb₁ nn]
+        refine NNReal.coe_sub ?_
+        refine div_le_self ?_ ?_
+        . exact zero_le_one' NNReal
+        . exact Nat.one_le_cast.mpr hn₀
+      rw [← hn₂]
+      refine hmo₀ n hn₀ ?_
+      refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+      refine lt_of_le_of_lt ?_ ha₀
+      refine hbr₃ _ ?_
+      rw [hsbr]
+      refine (Set.mem_image fr sb _).mpr ?_
+      use (fb nn)
+      rw [hfr, hsb₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self nn
+  . have hn₂: n + 1 ∈ sn := by
+      rw [hsn₀]
+      exact Set.mem_Ici.mpr (by linarith)
+    let nn : ↑sn := ⟨n + 1, hn₂⟩
+    have hn₃: f (n + 1) (fc (nn)) = 1 := by
+      rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+      exact rfl
+    rw [← hn₃]
+    refine hmo₀ (n + 1) (by linarith) ?_
+    refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+    refine lt_of_lt_of_le ha₁ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_4
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (a : ℝ)
+  (ha₀ : br < a)
+  (ha₁ : a < cr)
+  (ha₂ : 0 < a)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn):
+   f n a.toNNReal < f (n + 1) a.toNNReal ∧ f (n + 1) a.toNNReal < 1 := by
+  constructor
+  . refine h₈ n a.toNNReal hn₀ ?_ ?_
+    . exact Real.toNNReal_pos.mpr ha₂
+    . let nn : ↑sn := ⟨n, hn₁⟩
+      have hn₂: f n (fb nn) = 1 - 1 / n := by
+        rw [hf₁ n _ hn₀, hfb₁ nn]
+        refine NNReal.coe_sub ?_
+        refine div_le_self ?_ ?_
+        . exact zero_le_one' NNReal
+        . exact Nat.one_le_cast.mpr hn₀
+      rw [← hn₂]
+      refine hmo₀ n hn₀ ?_
+      refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+      refine lt_of_le_of_lt ?_ ha₀
+      refine hbr₃ _ ?_
+      rw [hsbr]
+      refine (Set.mem_image fr sb _).mpr ?_
+      use (fb nn)
+      rw [hfr, hsb₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self nn
+  . have hn₂: n + 1 ∈ sn := by
+      rw [hsn₀]
+      exact Set.mem_Ici.mpr (by linarith)
+    let nn : ↑sn := ⟨n + 1, hn₂⟩
+    have hn₃: f (n + 1) (fc (nn)) = 1 := by
+      rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+      exact rfl
+    rw [← hn₃]
+    refine hmo₀ (n + 1) (by linarith) ?_
+    refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+    refine lt_of_lt_of_le ha₁ ?_
+    refine hcr₃ _ ?_
+    rw [hscr]
+    refine (Set.mem_image fr sc _).mpr ?_
+    use (fc nn)
+    rw [hfr, hsc₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+
+
+lemma imo_1985_p6_exists_5
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (a : ℝ)
+  (ha₀ : br < a)
+  (ha₂ : 0 < a)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn):
+   f n a.toNNReal < f (n + 1) a.toNNReal := by
+  refine h₈ n a.toNNReal hn₀ ?_ ?_
+  . exact Real.toNNReal_pos.mpr ha₂
+  . let nn : ↑sn := ⟨n, hn₁⟩
+    have hn₂: f n (fb nn) = 1 - 1 / n := by
+      rw [hf₁ n _ hn₀, hfb₁ nn]
+      refine NNReal.coe_sub ?_
+      refine div_le_self ?_ ?_
+      . exact zero_le_one' NNReal
+      . exact Nat.one_le_cast.mpr hn₀
+    rw [← hn₂]
+    refine hmo₀ n hn₀ ?_
+    refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+    refine lt_of_le_of_lt ?_ ha₀
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    use (fb nn)
+    rw [hfr, hsb₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_6
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (a : ℝ)
+  (ha₀ : br < a)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn):
+   1 - 1 / ↑n < f n a.toNNReal := by
+  let nn : ↑sn := ⟨n, hn₁⟩
+  have hn₂: f n (fb nn) = 1 - 1 / n := by
+    rw [hf₁ n _ hn₀, hfb₁ nn]
+    refine NNReal.coe_sub ?_
+    refine div_le_self ?_ ?_
+    . exact zero_le_one' NNReal
+    . exact Nat.one_le_cast.mpr hn₀
+  rw [← hn₂]
+  refine hmo₀ n hn₀ ?_
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  refine lt_of_le_of_lt ?_ ha₀
+  refine hbr₃ _ ?_
+  rw [hsbr]
+  refine (Set.mem_image fr sb _).mpr ?_
+  use (fb nn)
+  rw [hfr, hsb₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_7
+  (f : ℕ → NNReal → ℝ)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n, hn₁⟩):
+  f n (fb nn) = 1 - 1 / ↑n := by
+  rw [hf₁ n _ hn₀, hnn, hfb₁ ⟨n, hn₁⟩]
+  refine NNReal.coe_sub ?_
+  refine div_le_self ?_ ?_
+  . exact zero_le_one' NNReal
+  . exact Nat.one_le_cast.mpr hn₀
+
+lemma imo_1985_p6_exists_8
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (a : ℝ)
+  (ha₀ : br < a)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (nn : ↑sn):
+   f n (fb nn) < f n a.toNNReal := by
+  refine hmo₀ n hn₀ ?_
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  refine lt_of_le_of_lt ?_ ha₀
+  refine hbr₃ _ ?_
+  rw [hsbr]
+  refine (Set.mem_image fr sb _).mpr ?_
+  use (fb nn)
+  rw [hfr, hsb₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_9
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (a : ℝ)
+  (ha₀ : br < a)
+  (nn : ↑sn):
+   fb nn < a.toNNReal := by
+  refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+  refine lt_of_le_of_lt ?_ ha₀
+  refine hbr₃ _ ?_
+  rw [hsbr]
+  refine (Set.mem_image fr sb _).mpr ?_
+  use (fb nn)
+  rw [hfr, hsb₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_10
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (a : ℝ)
+  (ha₀ : br < a)
+  (nn : ↑sn):
+   ↑(fb nn) < a := by
+  refine lt_of_le_of_lt ?_ ha₀
+  refine hbr₃ _ ?_
+  rw [hsbr]
+  refine (Set.mem_image fr sb _).mpr ?_
+  use (fb nn)
+  rw [hfr, hsb₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+
+lemma imo_1985_p6_exists_11
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (nn : ↑sn):
+   ∃ x ∈ sb, fr x = ↑(fb nn) := by
+  use (fb nn)
+  rw [hfr, hsb₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_12
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fc : ↑sn → NNReal)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (a : ℝ)
+  (ha₁ : a < cr)
+  (ha₂ : 0 < a)
+  (n : ℕ)
+  (hn₀ : 0 < n):
+   f (n + 1) a.toNNReal < 1 := by
+  have hn₂: n + 1 ∈ sn := by
+    rw [hsn₀]
+    exact Set.mem_Ici.mpr (by linarith)
+  let nn : ↑sn := ⟨n + 1, hn₂⟩
+  have hn₃: f (n + 1) (fc (nn)) = 1 := by
+    rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+    exact rfl
+  rw [← hn₃]
+  refine hmo₀ (n + 1) (by linarith) ?_
+  refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+  refine lt_of_lt_of_le ha₁ ?_
+  refine hcr₃ _ ?_
+  rw [hscr]
+  refine (Set.mem_image fr sc _).mpr ?_
+  use (fc nn)
+  rw [hfr, hsc₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_13
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (a : ℝ)
+  (ha₁ : a < cr)
+  (ha₂ : 0 < a)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n + 1, hn₂⟩):
+  f (n + 1) a.toNNReal < 1 := by
+  have hn₃: f (n + 1) (fc (nn)) = 1 := by
+    rw [hf₁ (n + 1) _ (by linarith), hnn, hfc₁ ⟨n + 1, hn₂⟩]
+    exact rfl
+  rw [← hn₃]
+  refine hmo₀ (n + 1) (by linarith) ?_
+  refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+  refine lt_of_lt_of_le ha₁ ?_
+  refine hcr₃ _ ?_
+  rw [hscr]
+  refine (Set.mem_image fr sc _).mpr ?_
+  use (fc nn)
+  rw [hfr, hsc₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+
+lemma imo_1985_p6_exists_14
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (a : ℝ)
+  (ha₁ : a < cr)
+  (ha₂ : 0 < a)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn := ⟨n + 1, hn₂⟩):
+   f (n + 1) a.toNNReal < f (n + 1) (fc nn) := by
+  refine hmo₀ (n + 1) (by linarith) ?_
+  refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+  refine lt_of_lt_of_le ha₁ ?_
+  refine hcr₃ _ ?_
+  rw [hscr]
+  refine (Set.mem_image fr sc _).mpr ?_
+  use (fc nn)
+  rw [hfr, hsc₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+
+lemma imo_1985_p6_exists_15
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (a : ℝ)
+  (ha₁ : a < cr)
+  (ha₂ : 0 < a)
+  (n : ℕ)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn := ⟨n + 1, hn₂⟩):
+   a.toNNReal < fc nn := by
+  refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+  refine lt_of_lt_of_le ha₁ ?_
+  refine hcr₃ _ ?_
+  rw [hscr]
+  refine (Set.mem_image fr sc _).mpr ?_
+  use (fc nn)
+  rw [hfr, hsc₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+
+lemma imo_1985_p6_exists_16
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (scr : Set ℝ)
+  (hscr : scr = fr '' sc)
+  (cr : ℝ)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (n : ℕ)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn := ⟨n + 1, hn₂⟩):
+   cr ≤ ↑(fc nn) := by
+  refine hcr₃ _ ?_
+  rw [hscr]
+  refine (Set.mem_image fr sc _).mpr ?_
+  use (fc nn)
+  rw [hfr, hsc₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+
+lemma imo_1985_p6_exists_17
+  (sn : Set ℕ)
+  (fc : ↑sn → NNReal)
+  (sc : Set NNReal)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (n : ℕ)
+  (hn₂ : n + 1 ∈ sn)
+  (nn : ↑sn := ⟨n + 1, hn₂⟩):
+   ∃ x ∈ sc, fr x = ↑(fc nn) := by
+  use (fc nn)
+  rw [hfr, hsc₀]
+  refine ⟨?_, rfl⟩
+  exact Set.mem_range_self nn
+
+lemma imo_1985_p6_exists_18
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hu₅ : br ≤ cr)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (hu₆ : br = cr):
+   ∃ x, ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1 := by
+  use br.toNNReal
+  intros n hn₀
+  have hn₁: n ∈ sn := by
+    rw [hsn₀]
+    exact hn₀
+  constructor
+  . refine h₂ n br.toNNReal ⟨hn₀, ?_⟩
+    exact Real.toNNReal_pos.mpr hbr₁
+  constructor
+  . refine h₈ n br.toNNReal hn₀ ?_ ?_
+    . exact Real.toNNReal_pos.mpr hbr₁
+    . let nn : ↑sn := ⟨n, hn₁⟩
+      have hn₂: fb nn < br := by
+        by_contra! hc₀
+        have hbr₅: (fb nn) = br := by
+          refine eq_of_le_of_le ?_ hc₀
+          refine hbr₃ _ ?_
+          rw [hsbr]
+          refine (Set.mem_image fr sb _).mpr ?_
+          use (fb nn)
+          rw [hfr, hsb₀]
+          constructor
+          . exact Set.mem_range_self nn
+          . exact rfl
+        have hn₂: n + 1 ∈ sn := by
+          rw [hsn₀]
+          refine Set.mem_Ici.mpr ?_
+          exact Nat.le_add_right_of_le hn₀
+        let ns : ↑sn := ⟨n + 1, hn₂⟩
+        have hc₁: fb nn < fb ns := by
+          refine hfb₃ ?_
+          refine Subtype.mk_lt_mk.mpr ?_
+          exact lt_add_one n
+        have hbr₆: fb ns ≤ fb nn := by
+          refine NNReal.coe_le_coe.mp ?_
+          rw [hbr₅]
+          refine hbr₃ _ ?_
+          rw [hsbr]
+          refine (Set.mem_image fr sb _).mpr ?_
+          use (fb ns)
+          rw [hfr, hsb₀]
+          refine ⟨?_, rfl⟩
+          exact Set.mem_range_self ns
+        refine (lt_self_iff_false (fb nn)).mp ?_
+        exact lt_of_lt_of_le hc₁ hbr₆
+      have hn₃: f n (fb nn) = 1 - 1 / n := by
+        rw [hf₁ n _ hn₀, hfb₁ nn]
+        refine NNReal.coe_sub ?_
+        refine div_le_self ?_ ?_
+        . exact zero_le_one' NNReal
+        . exact Nat.one_le_cast.mpr hn₀
+      rw [← hn₃]
+      refine hmo₀ n hn₀ ?_
+      exact Real.lt_toNNReal_iff_coe_lt.mpr hn₂
+  . have hn₂: n + 1 ∈ sn := by
+      rw [hsn₀]
+      exact Set.mem_Ici.mpr (by linarith)
+    let nn : ↑sn := ⟨n + 1, hn₂⟩
+    have hcr₁: 0 < cr := by exact gt_of_ge_of_gt hu₅ hbr₁
+    have hn₃: f (n + 1) (fc (nn)) = 1 := by
+      rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+      exact rfl
+    rw [← hn₃, hu₆]
+    refine hmo₀ (n + 1) (by linarith) ?_
+    refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt hcr₁)).mpr ?_
+    by_contra! hc₀
+    have hc₁: fc nn = cr := by
+      refine eq_of_le_of_le hc₀ ?_
+      refine hcr₃ _ ?_
+      rw [hscr]
+      refine (Set.mem_image fr sc _).mpr ?_
+      use (fc nn)
+      rw [hfr, hsc₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self nn
+    have hn₄: n + 2 ∈ sn := by
+      rw [hsn₀]
+      refine Set.mem_Ici.mpr ?_
+      exact Nat.le_add_right_of_le hn₀
+    let ns : ↑sn := ⟨n + 2, hn₄⟩
+    have hn₅: fc ns < fc nn := by
+      refine hfc₃ ?_
+      refine Subtype.mk_lt_mk.mpr ?_
+      exact Nat.lt_add_one (n + 1)
+    have hc₂: fc nn ≤ fc ns := by
+      refine NNReal.coe_le_coe.mp ?_
+      rw [hc₁]
+      refine hcr₃ _ ?_
+      rw [hscr]
+      refine (Set.mem_image fr sc _).mpr ?_
+      use (fc ns)
+      rw [hfr, hsc₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self ns
+    refine (lt_self_iff_false (fc ns)).mp ?_
+    exact lt_of_lt_of_le hn₅ hc₂
+
+
+lemma imo_1985_p6_exists_19
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hu₅ : br ≤ cr)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (hu₆ : br = cr)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn):
+   0 < f n br.toNNReal ∧ f n br.toNNReal < f (n + 1) br.toNNReal ∧ f (n + 1) br.toNNReal < 1 := by
+  constructor
+  . refine h₂ n br.toNNReal ⟨hn₀, ?_⟩
+    exact Real.toNNReal_pos.mpr hbr₁
+  constructor
+  . refine h₈ n br.toNNReal hn₀ ?_ ?_
+    . exact Real.toNNReal_pos.mpr hbr₁
+    . let nn : ↑sn := ⟨n, hn₁⟩
+      have hn₂: fb nn < br := by
+        by_contra! hc₀
+        have hbr₅: (fb nn) = br := by
+          refine eq_of_le_of_le ?_ hc₀
+          refine hbr₃ _ ?_
+          rw [hsbr]
+          refine (Set.mem_image fr sb _).mpr ?_
+          use (fb nn)
+          rw [hfr, hsb₀]
+          constructor
+          . exact Set.mem_range_self nn
+          . exact rfl
+        have hn₂: n + 1 ∈ sn := by
+          rw [hsn₀]
+          refine Set.mem_Ici.mpr ?_
+          exact Nat.le_add_right_of_le hn₀
+        let ns : ↑sn := ⟨n + 1, hn₂⟩
+        have hc₁: fb nn < fb ns := by
+          refine hfb₃ ?_
+          refine Subtype.mk_lt_mk.mpr ?_
+          exact lt_add_one n
+        have hbr₆: fb ns ≤ fb nn := by
+          refine NNReal.coe_le_coe.mp ?_
+          rw [hbr₅]
+          refine hbr₃ _ ?_
+          rw [hsbr]
+          refine (Set.mem_image fr sb _).mpr ?_
+          use (fb ns)
+          rw [hfr, hsb₀]
+          refine ⟨?_, rfl⟩
+          exact Set.mem_range_self ns
+        refine (lt_self_iff_false (fb nn)).mp ?_
+        exact lt_of_lt_of_le hc₁ hbr₆
+      have hn₃: f n (fb nn) = 1 - 1 / n := by
+        rw [hf₁ n _ hn₀, hfb₁ nn]
+        refine NNReal.coe_sub ?_
+        refine div_le_self ?_ ?_
+        . exact zero_le_one' NNReal
+        . exact Nat.one_le_cast.mpr hn₀
+      rw [← hn₃]
+      refine hmo₀ n hn₀ ?_
+      exact Real.lt_toNNReal_iff_coe_lt.mpr hn₂
+  . have hn₂: n + 1 ∈ sn := by
+      rw [hsn₀]
+      exact Set.mem_Ici.mpr (by linarith)
+    let nn : ↑sn := ⟨n + 1, hn₂⟩
+    have hcr₁: 0 < cr := by exact gt_of_ge_of_gt hu₅ hbr₁
+    have hn₃: f (n + 1) (fc (nn)) = 1 := by
+      rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+      exact rfl
+    rw [← hn₃, hu₆]
+    refine hmo₀ (n + 1) (by linarith) ?_
+    refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt hcr₁)).mpr ?_
+    by_contra! hc₀
+    have hc₁: fc nn = cr := by
+      refine eq_of_le_of_le hc₀ ?_
+      refine hcr₃ _ ?_
+      rw [hscr]
+      refine (Set.mem_image fr sc _).mpr ?_
+      use (fc nn)
+      rw [hfr, hsc₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self nn
+    have hn₄: n + 2 ∈ sn := by
+      rw [hsn₀]
+      refine Set.mem_Ici.mpr ?_
+      exact Nat.le_add_right_of_le hn₀
+    let ns : ↑sn := ⟨n + 2, hn₄⟩
+    have hn₅: fc ns < fc nn := by
+      refine hfc₃ ?_
+      refine Subtype.mk_lt_mk.mpr ?_
+      exact Nat.lt_add_one (n + 1)
+    have hc₂: fc nn ≤ fc ns := by
+      refine NNReal.coe_le_coe.mp ?_
+      rw [hc₁]
+      refine hcr₃ _ ?_
+      rw [hscr]
+      refine (Set.mem_image fr sc _).mpr ?_
+      use (fc ns)
+      rw [hfr, hsc₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self ns
+    refine (lt_self_iff_false (fc ns)).mp ?_
+    exact lt_of_lt_of_le hn₅ hc₂
+
+lemma imo_1985_p6_exists_20
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hu₅ : br ≤ cr)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x)
+  (hu₆ : br = cr)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn):
+   f n br.toNNReal < f (n + 1) br.toNNReal ∧ f (n + 1) br.toNNReal < 1 := by
+  constructor
+  . refine h₈ n br.toNNReal hn₀ ?_ ?_
+    . exact Real.toNNReal_pos.mpr hbr₁
+    . let nn : ↑sn := ⟨n, hn₁⟩
+      have hn₂: fb nn < br := by
+        by_contra! hc₀
+        have hbr₅: (fb nn) = br := by
+          refine eq_of_le_of_le ?_ hc₀
+          refine hbr₃ _ ?_
+          rw [hsbr]
+          refine (Set.mem_image fr sb _).mpr ?_
+          use (fb nn)
+          rw [hfr, hsb₀]
+          constructor
+          . exact Set.mem_range_self nn
+          . exact rfl
+        have hn₂: n + 1 ∈ sn := by
+          rw [hsn₀]
+          refine Set.mem_Ici.mpr ?_
+          exact Nat.le_add_right_of_le hn₀
+        let ns : ↑sn := ⟨n + 1, hn₂⟩
+        have hc₁: fb nn < fb ns := by
+          refine hfb₃ ?_
+          refine Subtype.mk_lt_mk.mpr ?_
+          exact lt_add_one n
+        have hbr₆: fb ns ≤ fb nn := by
+          refine NNReal.coe_le_coe.mp ?_
+          rw [hbr₅]
+          refine hbr₃ _ ?_
+          rw [hsbr]
+          refine (Set.mem_image fr sb _).mpr ?_
+          use (fb ns)
+          rw [hfr, hsb₀]
+          refine ⟨?_, rfl⟩
+          exact Set.mem_range_self ns
+        refine (lt_self_iff_false (fb nn)).mp ?_
+        exact lt_of_lt_of_le hc₁ hbr₆
+      have hn₃: f n (fb nn) = 1 - 1 / n := by
+        rw [hf₁ n _ hn₀, hfb₁ nn]
+        refine NNReal.coe_sub ?_
+        refine div_le_self ?_ ?_
+        . exact zero_le_one' NNReal
+        . exact Nat.one_le_cast.mpr hn₀
+      rw [← hn₃]
+      refine hmo₀ n hn₀ ?_
+      exact Real.lt_toNNReal_iff_coe_lt.mpr hn₂
+  . have hn₂: n + 1 ∈ sn := by
+      rw [hsn₀]
+      exact Set.mem_Ici.mpr (by linarith)
+    let nn : ↑sn := ⟨n + 1, hn₂⟩
+    have hcr₁: 0 < cr := by exact gt_of_ge_of_gt hu₅ hbr₁
+    have hn₃: f (n + 1) (fc (nn)) = 1 := by
+      rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+      exact rfl
+    rw [← hn₃, hu₆]
+    refine hmo₀ (n + 1) (by linarith) ?_
+    refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt hcr₁)).mpr ?_
+    by_contra! hc₀
+    have hc₁: fc nn = cr := by
+      refine eq_of_le_of_le hc₀ ?_
+      refine hcr₃ _ ?_
+      rw [hscr]
+      refine (Set.mem_image fr sc _).mpr ?_
+      use (fc nn)
+      rw [hfr, hsc₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self nn
+    have hn₄: n + 2 ∈ sn := by
+      rw [hsn₀]
+      refine Set.mem_Ici.mpr ?_
+      exact Nat.le_add_right_of_le hn₀
+    let ns : ↑sn := ⟨n + 2, hn₄⟩
+    have hn₅: fc ns < fc nn := by
+      refine hfc₃ ?_
+      refine Subtype.mk_lt_mk.mpr ?_
+      exact Nat.lt_add_one (n + 1)
+    have hc₂: fc nn ≤ fc ns := by
+      refine NNReal.coe_le_coe.mp ?_
+      rw [hc₁]
+      refine hcr₃ _ ?_
+      rw [hscr]
+      refine (Set.mem_image fr sc _).mpr ?_
+      use (fc ns)
+      rw [hfr, hsc₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self ns
+    refine (lt_self_iff_false (fc ns)).mp ?_
+    exact lt_of_lt_of_le hn₅ hc₂
+
+
+
+
+lemma imo_1985_p6_exists_21
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfb₃ : StrictMono fb)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn):
+   f n br.toNNReal < f (n + 1) br.toNNReal := by
+  refine h₈ n br.toNNReal hn₀ ?_ ?_
+  . exact Real.toNNReal_pos.mpr hbr₁
+  . let nn : ↑sn := ⟨n, hn₁⟩
+    have hn₂: fb nn < br := by
+      by_contra! hc₀
+      have hbr₅: (fb nn) = br := by
+        refine eq_of_le_of_le ?_ hc₀
+        refine hbr₃ _ ?_
+        rw [hsbr]
+        refine (Set.mem_image fr sb _).mpr ?_
+        use (fb nn)
+        rw [hfr, hsb₀]
+        constructor
+        . exact Set.mem_range_self nn
+        . exact rfl
+      have hn₂: n + 1 ∈ sn := by
+        rw [hsn₀]
+        refine Set.mem_Ici.mpr ?_
+        exact Nat.le_add_right_of_le hn₀
+      let ns : ↑sn := ⟨n + 1, hn₂⟩
+      have hc₁: fb nn < fb ns := by
+        refine hfb₃ ?_
+        refine Subtype.mk_lt_mk.mpr ?_
+        exact lt_add_one n
+      have hbr₆: fb ns ≤ fb nn := by
+        refine NNReal.coe_le_coe.mp ?_
+        rw [hbr₅]
+        refine hbr₃ _ ?_
+        rw [hsbr]
+        refine (Set.mem_image fr sb _).mpr ?_
+        use (fb ns)
+        rw [hfr, hsb₀]
+        refine ⟨?_, rfl⟩
+        exact Set.mem_range_self ns
+      refine (lt_self_iff_false (fb nn)).mp ?_
+      exact lt_of_lt_of_le hc₁ hbr₆
+    have hn₃: f n (fb nn) = 1 - 1 / n := by
+      rw [hf₁ n _ hn₀, hfb₁ nn]
+      refine NNReal.coe_sub ?_
+      refine div_le_self ?_ ?_
+      . exact zero_le_one' NNReal
+      . exact Nat.one_le_cast.mpr hn₀
+    rw [← hn₃]
+    refine hmo₀ n hn₀ ?_
+    exact Real.lt_toNNReal_iff_coe_lt.mpr hn₂
+
+
+lemma imo_1985_p6_exists_22
+  (f : ℕ → NNReal → ℝ)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfb₃ : StrictMono fb)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n, hn₁⟩):
+   1 - 1 / ↑n < f n br.toNNReal := by
+  have hn₂: fb nn < br := by
+    by_contra! hc₀
+    have hbr₅: (fb nn) = br := by
+      refine eq_of_le_of_le ?_ hc₀
+      refine hbr₃ _ ?_
+      rw [hsbr]
+      refine (Set.mem_image fr sb _).mpr ?_
+      use (fb nn)
+      rw [hfr, hsb₀]
+      constructor
+      . exact Set.mem_range_self nn
+      . exact rfl
+    have hn₂: n + 1 ∈ sn := by
+      rw [hsn₀]
+      refine Set.mem_Ici.mpr ?_
+      exact Nat.le_add_right_of_le hn₀
+    let ns : ↑sn := ⟨n + 1, hn₂⟩
+    have hc₁: fb nn < fb ns := by
+      refine hfb₃ ?_
+      refine Subtype.mk_lt_mk.mpr ?_
+      rw [hnn]
+      exact lt_add_one n
+    have hbr₆: fb ns ≤ fb nn := by
+      refine NNReal.coe_le_coe.mp ?_
+      rw [hbr₅]
+      refine hbr₃ _ ?_
+      rw [hsbr]
+      refine (Set.mem_image fr sb _).mpr ?_
+      use (fb ns)
+      rw [hfr, hsb₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self ns
+    refine (lt_self_iff_false (fb nn)).mp ?_
+    exact lt_of_lt_of_le hc₁ hbr₆
+  have hn₃: f n (fb nn) = 1 - 1 / n := by
+    rw [hf₁ n _ hn₀, hnn, hfb₁ ⟨n, hn₁⟩]
+    refine NNReal.coe_sub ?_
+    refine div_le_self ?_ ?_
+    . exact zero_le_one' NNReal
+    . exact Nat.one_le_cast.mpr hn₀
+  rw [← hn₃]
+  refine hmo₀ n hn₀ ?_
+  exact Real.lt_toNNReal_iff_coe_lt.mpr hn₂
+
+
+lemma imo_1985_p6_exists_23
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n, hn₁⟩):
+   ↑(fb nn) < br := by
+  by_contra! hc₀
+  have hbr₅: (fb nn) = br := by
+    refine eq_of_le_of_le ?_ hc₀
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    use (fb nn)
+    rw [hfr, hsb₀]
+    constructor
+    . exact Set.mem_range_self nn
+    . exact rfl
+  have hn₂: n + 1 ∈ sn := by
+    rw [hsn₀]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  let ns : ↑sn := ⟨n + 1, hn₂⟩
+  have hc₁: fb nn < fb ns := by
+    refine hfb₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hnn]
+    exact lt_add_one n
+  have hbr₆: fb ns ≤ fb nn := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hbr₅]
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    use (fb ns)
+    rw [hfr, hsb₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fb nn)).mp ?_
+  exact lt_of_lt_of_le hc₁ hbr₆
+
+lemma imo_1985_p6_exists_24
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb : ↑sn → NNReal)
+  (hfb₃ : StrictMono fb)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (n : ℕ)
+  (hn₀ : 0 < n)
+  (hn₁ : n ∈ sn)
+  (nn : ↑sn)
+  (hnn : nn = ⟨n, hn₁⟩)
+  (hc₀ : br ≤ ↑(fb nn)):
+   False := by
+  have hbr₅: (fb nn) = br := by
+    refine eq_of_le_of_le ?_ hc₀
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    use (fb nn)
+    rw [hfr, hsb₀]
+    constructor
+    . exact Set.mem_range_self nn
+    . exact rfl
+  have hn₂: n + 1 ∈ sn := by
+    rw [hsn₀]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right_of_le hn₀
+  let ns : ↑sn := ⟨n + 1, hn₂⟩
+  have hc₁: fb nn < fb ns := by
+    refine hfb₃ ?_
+    refine Subtype.mk_lt_mk.mpr ?_
+    rw [hnn]
+    exact lt_add_one n
+  have hbr₆: fb ns ≤ fb nn := by
+    refine NNReal.coe_le_coe.mp ?_
+    rw [hbr₅]
+    refine hbr₃ _ ?_
+    rw [hsbr]
+    refine (Set.mem_image fr sb _).mpr ?_
+    use (fb ns)
+    rw [hfr, hsb₀]
+    refine ⟨?_, rfl⟩
+    exact Set.mem_range_self ns
+  refine (lt_self_iff_false (fb nn)).mp ?_
+  exact lt_of_lt_of_le hc₁ hbr₆
+
+lemma imo_1985_p6_exists_25
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (nn : ↑sn)
+  (hc₀ : br ≤ ↑(fb nn)):
+   ↑(fb nn) = br := by
+  refine eq_of_le_of_le ?_ hc₀
+  refine hbr₃ _ ?_
+  rw [hsbr]
+  refine (Set.mem_image fr sb _).mpr ?_
+  use (fb nn)
+  rw [hfr, hsb₀]
+  constructor
+  . exact Set.mem_range_self nn
+  . exact rfl
+
+lemma imo_1985_p6_exists_26
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (sb : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (nn : ↑sn):
+   ↑(fb nn) ≤ br := by
+  refine hbr₃ _ ?_
+  rw [hsbr]
+  refine (Set.mem_image fr sb _).mpr ?_
+  use (fb nn)
+  rw [hfr, hsb₀]
+  constructor
+  . exact Set.mem_range_self nn
+  . exact rfl
+
+
+lemma imo_1985_p6_8_14
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₁ : 1 < n.1)
+  (g₀ : ↑↑n ≠ (0:NNReal)):
+   (↑↑n - 1) / ↑↑n ≠ (0:NNReal) := by
+  refine div_ne_zero ?_ g₀
+  norm_cast
+  exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+
+lemma imo_1985_p6_8_15
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hz₁ : (f (↑n) z).toNNReal = 1 - 1 / ↑↑n)
+  (hz₂ : 1 - 1 / ↑↑n ≠ (0:NNReal)):
+   f (↑n) z = 1 - 1 / ↑↑n := by
+  apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+  rw [hz₁]
+  exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+
+
+lemma imo_1985_p6_8_16
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hz₁ : (f (↑n) z).toNNReal = 1 - 1 / ↑↑n)
+  (hn₁ : ¬ (1:ℕ) < ↑n):
+  f (↑n) z = 1 - 1 / ↑↑n := by
+  have hn₂: n.1 = 1 := by linarith
+  rw [hn₂, h₀] at hz₁
+  simp at hz₁
+  rw [hn₂, h₀, hz₁]
+  simp
+
+lemma imo_1985_p6_8_17
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (z : NNReal)
+  (hz₁ : (↑z:ℝ).toNNReal = 1 - 1 / ↑1)
+  (hn₂ : ↑n.1 = 1):
+   f (↑n) z = 1 - 1 / ↑↑n := by
+  simp at hz₁
+  rw [hn₂, h₀, hz₁]
+  simp
+
+
+
+lemma imo_1985_p6_8_18
+  (f : ℕ → NNReal → ℝ)
+  (f₀ : ℕ → NNReal → NNReal)
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hc₁ : 1 ≤ f (↑n) z)
+  (hz₁ : f₀ (↑n) z = 1 - 1 / ↑↑n)
+  (hz₃ : f (↑n) z = 1 - 1 / ↑↑n):
+   False := by
+  rw [hz₃] at hc₁
+  have hz₄: 0 < 1 / (n:ℝ) := by
+    refine div_pos (by linarith) ?_
+    exact Nat.cast_pos'.mpr hn₀
+  linarith
+
+lemma imo_1985_p6_8_19
+  (f : ℕ → NNReal → ℝ)
+  (f₀ : ℕ → NNReal → NNReal)
+  (sn : Set ℕ)
+  (n : ↑sn)
+  (hn₀ : 0 < n.1)
+  (z : NNReal)
+  (hc₁ : 1 ≤ f (↑n) z)
+  (hz₁ : f₀ (↑n) z = 1 - 1 / ↑↑n)
+  (hz₃ : f (↑n) z = 1 - 1 / ↑↑n):
+   False := by
+  rw [hz₃] at hc₁
+  have hz₄: 0 < 1 / (n:ℝ) := by
+    refine div_pos (by linarith) ?_
+    exact Nat.cast_pos'.mpr hn₀
+  linarith
+
+
+lemma imo_1985_p6_9_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hf₅ : ∀ (x : NNReal), fi 1 x = x)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (fb : ℕ → NNReal)
+  (hfb₀ : fb = fun n => fi n (1 - 1 / ↑n))
+  (m : ℕ)
+  (hm₀ : 1 < m):
+   fb (Order.pred m) < fb m := by
+  rw [hfb₀]
+  refine Nat.le_induction ?_ ?_ m hm₀
+  . have g₁: fi 1 0 = 0 := by exact hf₅ 0
+    have g₂: (2:NNReal).IsConjExponent (2:NNReal) := by
+      refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+      . exact one_lt_two
+      . norm_cast
+        simp
+    simp
+    norm_cast
+    rw [g₁, NNReal.IsConjExponent.one_sub_inv g₂]
+    let x := fi 2 2⁻¹
+    have hx₀: x = fi 2 2⁻¹ := by rfl
+    have hx₁: f₀ 2 x = 2⁻¹ := by
+      rw [hx₀]
+      have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+      exact g₃ 2⁻¹
+    rw [← hx₀]
+    contrapose! hx₁
+    have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+    have hc₃: f₀ 2 x = 0 := by
+      rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+      norm_cast
+      rw [zero_mul]
+      exact Real.toNNReal_zero
+    rw [hc₃]
+    exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+  . simp
+    intros n hn₀ _
+    let i := fi n (1 - (↑n)⁻¹)
+    let j := fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹)
+    have hi₀: i = fi n (1 - (↑n)⁻¹) := by rfl
+    have hj₀: j = fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹) := by rfl
+    have hi₁: f₀ n i = (1 - (↑n)⁻¹) := by exact (hf₇ n i (1 - (↑n:NNReal)⁻¹) (by linarith)).mpr hi₀.symm
+    have hj₁: f₀ (n + 1) j = (1 - ((↑n:NNReal) + 1)⁻¹) := by
+      exact (hf₇ (n + 1) j _ (by linarith)).mpr hj₀.symm
+    have hj₂: (1 - ((↑n:NNReal) + 1)⁻¹) = (1 - ((n:ℝ) + 1)⁻¹).toNNReal := by
+      exact rfl
+    have hn₂: f₀ (n + 1) i < f₀ (n + 1) j := by
+      rw [hj₁, hj₂, hf₂ (n + 1) _ (by linarith), h₁ n i (by linarith)]
+      rw [hf₁ n i (by linarith), hi₁]
+      refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+      . refine sub_pos.mpr ?_
+        refine inv_lt_one_of_one_lt₀ ?_
+        norm_cast
+        exact Nat.lt_add_right 1 hn₀
+      . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+        rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+        simp
+        refine inv_strictAnti₀ ?_ ?_
+        . norm_cast
+          exact Nat.zero_lt_of_lt hn₀
+        . norm_cast
+          exact lt_add_one n
+    refine (StrictMono.lt_iff_lt ?_).mp hn₂
+    exact hmo₂ (n + 1) (by linarith)
+
+
+lemma imo_1985_p6_9_2
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hf₅ : ∀ (x : NNReal), fi 1 x = x)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (m : ℕ)
+  (hm₀ : 1 < m):
+   (fun n => fi n (1 - 1 / ↑n)) (Order.pred m) < (fun n => fi n (1 - 1 / ↑n)) m := by
+  refine Nat.le_induction ?_ ?_ m hm₀
+  . have g₁: fi 1 0 = 0 := by exact hf₅ 0
+    have g₂: (2:NNReal).IsConjExponent (2:NNReal) := by
+      refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+      . exact one_lt_two
+      . norm_cast
+        simp
+    simp
+    norm_cast
+    rw [g₁, NNReal.IsConjExponent.one_sub_inv g₂]
+    let x := fi 2 2⁻¹
+    have hx₀: x = fi 2 2⁻¹ := by rfl
+    have hx₁: f₀ 2 x = 2⁻¹ := by
+      rw [hx₀]
+      have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+      exact g₃ 2⁻¹
+    rw [← hx₀]
+    contrapose! hx₁
+    have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+    have hc₃: f₀ 2 x = 0 := by
+      rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+      norm_cast
+      rw [zero_mul]
+      exact Real.toNNReal_zero
+    rw [hc₃]
+    exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+  . simp
+    intros n hn₀ _
+    let i := fi n (1 - (↑n)⁻¹)
+    let j := fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹)
+    have hi₀: i = fi n (1 - (↑n)⁻¹) := by rfl
+    have hj₀: j = fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹) := by rfl
+    have hi₁: f₀ n i = (1 - (↑n)⁻¹) := by exact (hf₇ n i (1 - (↑n:NNReal)⁻¹) (by linarith)).mpr hi₀.symm
+    have hj₁: f₀ (n + 1) j = (1 - ((↑n:NNReal) + 1)⁻¹) := by
+      exact (hf₇ (n + 1) j _ (by linarith)).mpr hj₀.symm
+    have hj₂: (1 - ((↑n:NNReal) + 1)⁻¹) = (1 - ((n:ℝ) + 1)⁻¹).toNNReal := by
+      exact rfl
+    have hn₂: f₀ (n + 1) i < f₀ (n + 1) j := by
+      rw [hj₁, hj₂, hf₂ (n + 1) _ (by linarith), h₁ n i (by linarith)]
+      rw [hf₁ n i (by linarith), hi₁]
+      refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+      . refine sub_pos.mpr ?_
+        refine inv_lt_one_of_one_lt₀ ?_
+        norm_cast
+        exact Nat.lt_add_right 1 hn₀
+      . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+        rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+        simp
+        refine inv_strictAnti₀ ?_ ?_
+        . norm_cast
+          exact Nat.zero_lt_of_lt hn₀
+        . norm_cast
+          exact lt_add_one n
+    refine (StrictMono.lt_iff_lt ?_).mp hn₂
+    exact hmo₂ (n + 1) (by linarith)
+
+lemma imo_1985_p6_9_3
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hf₅ : ∀ (x : NNReal), fi 1 x = x)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n)):
+   (fun n => fi n (1 - 1 / ↑n)) (Order.pred (Nat.succ 1)) < (fun n => fi n (1 - 1 / ↑n)) (Nat.succ 1) := by
+  have g₁: fi 1 0 = 0 := by exact hf₅ 0
+  have g₂: (2:NNReal).IsConjExponent (2:NNReal) := by
+    refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+    . exact one_lt_two
+    . norm_cast
+      simp
+  simp
+  norm_cast
+  rw [g₁, NNReal.IsConjExponent.one_sub_inv g₂]
+  let x := fi 2 2⁻¹
+  have hx₀: x = fi 2 2⁻¹ := by rfl
+  have hx₁: f₀ 2 x = 2⁻¹ := by
+    rw [hx₀]
+    have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+    exact g₃ 2⁻¹
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+
+
+lemma imo_1985_p6_9_4
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (g₁ : fi 1 0 = 0):
+   (fun n => fi n (1 - 1 / ↑n)) (Order.pred (Nat.succ 1)) < (fun n => fi n (1 - 1 / ↑n)) (Nat.succ 1) := by
+  have g₂: (2:NNReal).IsConjExponent (2:NNReal) := by
+    refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+    . exact one_lt_two
+    . norm_cast
+      simp
+  simp
+  norm_cast
+  rw [g₁, NNReal.IsConjExponent.one_sub_inv g₂]
+  let x := fi 2 2⁻¹
+  have hx₀: x = fi 2 2⁻¹ := by rfl
+  have hx₁: f₀ 2 x = 2⁻¹ := by
+    rw [hx₀]
+    have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+    exact g₃ 2⁻¹
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+
+lemma imo_1985_p6_9_5
+  (fi : ℕ → NNReal → NNReal)
+  (m : ℕ):
+   NNReal.IsConjExponent 2 2 := by
+  refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+  . exact one_lt_two
+  . norm_cast
+    simp
+
+lemma imo_1985_p6_9_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (g₁ : fi 1 0 = 0)
+  (g₂ : NNReal.IsConjExponent 2 2):
+   (fun n => fi n (1 - 1 / ↑n)) (Order.pred (Nat.succ 1)) < (fun n => fi n (1 - 1 / ↑n)) (Nat.succ 1) := by
+  simp
+  norm_cast
+  rw [g₁, NNReal.IsConjExponent.one_sub_inv g₂]
+  let x := fi 2 2⁻¹
+  have hx₀: x = fi 2 2⁻¹ := by rfl
+  have hx₁: f₀ 2 x = 2⁻¹ := by
+    rw [hx₀]
+    have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+    exact g₃ 2⁻¹
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+
+
+lemma imo_1985_p6_9_7
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (g₁ : fi 1 0 = 0)
+  (g₂ : NNReal.IsConjExponent 2 2):
+   fi 1 0 < fi 2 (1 - 2⁻¹) := by
+  rw [g₁, NNReal.IsConjExponent.one_sub_inv g₂]
+  let x := fi 2 2⁻¹
+  have hx₀: x = fi 2 2⁻¹ := by rfl
+  have hx₁: f₀ 2 x = 2⁻¹ := by
+    rw [hx₀]
+    have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+    exact g₃ 2⁻¹
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+
+
+lemma imo_1985_p6_9_8
+  (fi : ℕ → NNReal → NNReal)
+  (g₁ : fi 1 0 = 0)
+  (g₂ : NNReal.IsConjExponent 2 2)
+  (g₃ : 0 < fi 2 2⁻¹) :
+  (fun n ↦ fi n (1 - 1 / ↑n)) (Order.pred (Nat.succ 1)) < (fun n ↦ fi n (1 - 1 / ↑n)) (Nat.succ 1) := by
+  simp
+  norm_cast
+  rw [g₁, NNReal.IsConjExponent.one_sub_inv g₂]
+  exact g₃
+
+
+
+lemma imo_1985_p6_9_9
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (g₂ : NNReal.IsConjExponent 2 2):
+   0 < fi 2 2⁻¹ := by
+  let x := fi 2 2⁻¹
+  have hx₀: x = fi 2 2⁻¹ := by rfl
+  have hx₁: f₀ 2 x = 2⁻¹ := by
+    rw [hx₀]
+    have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+    exact g₃ 2⁻¹
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+
+
+lemma imo_1985_p6_9_10
+  (f₀ : ℕ → NNReal → NNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (x : NNReal := fi 2 2⁻¹)
+  (hx₀ : x = fi 2 2⁻¹):
+   f₀ 2 x = 2⁻¹ := by
+  rw [hx₀]
+  have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+  exact g₃ 2⁻¹
+
+
+lemma imo_1985_p6_9_11
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (g₂ : NNReal.IsConjExponent 2 2)
+  (x : NNReal := fi 2 2⁻¹)
+  (hx₀ : x = fi 2 2⁻¹)
+  (hx₁ : f₀ 2 x = 2⁻¹):
+   0 < fi 2 2⁻¹ := by
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+
+
+lemma imo_1985_p6_9_12
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (g₂ : NNReal.IsConjExponent 2 2)
+  (x : NNReal := fi 2 2⁻¹)
+  (hx₁ : x ≤ 0):
+   f₀ 2 x ≠ 2⁻¹ := by
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+
+
+lemma imo_1985_p6_9_13
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (g₂ : NNReal.IsConjExponent 2 2)
+  (x : NNReal := fi 2 2⁻¹)
+  (hc₁ : x = 0):
+   f₀ 2 x ≠ 2⁻¹ := by
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+
+
+lemma imo_1985_p6_9_14
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (x : NNReal := fi 2 2⁻¹)
+  (hc₁ : x = 0):
+   f₀ 2 x = 0 := by
+  rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+  norm_cast
+  rw [zero_mul]
+  exact Real.toNNReal_zero
+
+lemma imo_1985_p6_9_15
+  (f₀ : ℕ → NNReal → NNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (g₂ : NNReal.IsConjExponent 2 2)
+  (x : NNReal := fi 2 2⁻¹)
+  (hc₃ : f₀ 2 x = 0):
+   f₀ 2 x ≠ 2⁻¹ := by
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+
+lemma imo_1985_p6_9_16
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x)):
+  ∀ (n : ℕ), Nat.succ 1 ≤ n →
+    (fun n => fi n (1 - 1 / ↑n)) (Order.pred n) < (fun n => fi n (1 - 1 / ↑n)) n →
+      (fun n => fi n (1 - 1 / ↑n)) (Order.pred (n + 1)) < (fun n => fi n (1 - 1 / ↑n)) (n + 1) := by
+  simp
+  intros n hn₀ _
+  let i := fi n (1 - (↑n)⁻¹)
+  let j := fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹)
+  have hi₀: i = fi n (1 - (↑n)⁻¹) := by rfl
+  have hj₀: j = fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹) := by rfl
+  have hi₁: f₀ n i = (1 - (↑n)⁻¹) := by exact (hf₇ n i (1 - (↑n:NNReal)⁻¹) (by linarith)).mpr hi₀.symm
+  have hj₁: f₀ (n + 1) j = (1 - ((↑n:NNReal) + 1)⁻¹) := by
+    exact (hf₇ (n + 1) j _ (by linarith)).mpr hj₀.symm
+  have hj₂: (1 - ((↑n:NNReal) + 1)⁻¹) = (1 - ((n:ℝ) + 1)⁻¹).toNNReal := by
+    exact rfl
+  have hn₂: f₀ (n + 1) i < f₀ (n + 1) j := by
+    rw [hj₁, hj₂, hf₂ (n + 1) _ (by linarith), h₁ n i (by linarith)]
+    rw [hf₁ n i (by linarith), hi₁]
+    refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+    . refine sub_pos.mpr ?_
+      refine inv_lt_one_of_one_lt₀ ?_
+      norm_cast
+      exact Nat.lt_add_right 1 hn₀
+    . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+      rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+      simp
+      refine inv_strictAnti₀ ?_ ?_
+      . norm_cast
+        exact Nat.zero_lt_of_lt hn₀
+      . norm_cast
+        exact lt_add_one n
+  refine (StrictMono.lt_iff_lt ?_).mp hn₂
+  exact hmo₂ (n + 1) (by linarith)
+
+lemma imo_1985_p6_9_17
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (n : ℕ)
+  (hn₀ : 2 ≤ n) :
+   fi n (1 - (↑n)⁻¹) < fi (n + 1) (1 - (↑n + 1:NNReal)⁻¹) := by
+  let i := fi n (1 - (↑n)⁻¹)
+  let j := fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹)
+  have hi₀: i = fi n (1 - (↑n)⁻¹) := by rfl
+  have hj₀: j = fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹) := by rfl
+  have hi₁: f₀ n i = (1 - (↑n)⁻¹) := by exact (hf₇ n i (1 - (↑n:NNReal)⁻¹) (by linarith)).mpr hi₀.symm
+  have hj₁: f₀ (n + 1) j = (1 - ((↑n:NNReal) + 1)⁻¹) := by
+    exact (hf₇ (n + 1) j _ (by linarith)).mpr hj₀.symm
+  have hj₂: (1 - ((↑n:NNReal) + 1)⁻¹) = (1 - ((n:ℝ) + 1)⁻¹).toNNReal := by
+    exact rfl
+  have hn₂: f₀ (n + 1) i < f₀ (n + 1) j := by
+    rw [hj₁, hj₂, hf₂ (n + 1) _ (by linarith), h₁ n i (by linarith)]
+    rw [hf₁ n i (by linarith), hi₁]
+    refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+    . refine sub_pos.mpr ?_
+      refine inv_lt_one_of_one_lt₀ ?_
+      norm_cast
+      exact Nat.lt_add_right 1 hn₀
+    . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+      rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+      simp
+      refine inv_strictAnti₀ ?_ ?_
+      . norm_cast
+        exact Nat.zero_lt_of_lt hn₀
+      . norm_cast
+        exact lt_add_one n
+  refine (StrictMono.lt_iff_lt ?_).mp hn₂
+  exact hmo₂ (n + 1) (by linarith)
+
+
+lemma imo_1985_p6_9_18
+  (f : �� → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (i : NNReal := fi n (1 - (↑n)⁻¹))
+  (j : NNReal := fi (n + 1) (1 - (↑n + 1:NNReal)⁻¹))
+  (hi₀ : i = fi n (1 - (↑n)⁻¹))
+  (hj₀ : j = fi (n + 1) (1 - (↑n + 1:NNReal)⁻¹))
+  (hi₁ : f₀ n i = 1 - (↑n)⁻¹)
+  (hj₁ : f₀ (n + 1) j = 1 - (↑n + 1:NNReal)⁻¹)
+  (hj₂ : (1 - ((↑n:NNReal) + 1)⁻¹) = (1 - ((n:ℝ) + 1)⁻¹).toNNReal):
+  fi n (1 - (↑n)⁻¹) < fi (n + 1) (1 - (↑n + 1:NNReal)⁻¹) := by
+  have hn₂: f₀ (n + 1) i < f₀ (n + 1) j := by
+    rw [hj₁, hj₂, hf₂ (n + 1) _ (by linarith), h₁ n i (by linarith)]
+    rw [hf₁ n i (by linarith), hi₁]
+    refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+    . refine sub_pos.mpr ?_
+      refine inv_lt_one_of_one_lt₀ ?_
+      norm_cast
+      exact Nat.lt_add_right 1 hn₀
+    . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+      rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+      simp
+      refine inv_strictAnti₀ ?_ ?_
+      . norm_cast
+        exact Nat.zero_lt_of_lt hn₀
+      . norm_cast
+        exact lt_add_one n
+  rw [← hi₀, ← hj₀]
+  refine (StrictMono.lt_iff_lt ?_).mp hn₂
+  exact hmo₂ (n + 1) (by linarith)
+
+lemma imo_1985_p6_9_19
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (i j: NNReal)
+  (hi₁ : f₀ n i = 1 - (↑n)⁻¹)
+  (hj₁ : f₀ (n + 1) j = 1 - (↑n + 1:NNReal)⁻¹)
+  (hj₂ : (1 - ((↑n:NNReal) + 1)⁻¹) = (1 - ((n:ℝ) + 1)⁻¹).toNNReal):
+   f₀ (n + 1) i < f₀ (n + 1) j := by
+  rw [hj₁, hj₂, hf₂ (n + 1) _ (by linarith), h₁ n i (by linarith)]
+  rw [hf₁ n i (by linarith), hi₁]
+  refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+  . refine sub_pos.mpr ?_
+    refine inv_lt_one_of_one_lt₀ ?_
+    norm_cast
+    exact Nat.lt_add_right 1 hn₀
+  . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+    rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+    simp
+    refine inv_strictAnti₀ ?_ ?_
+    . norm_cast
+      exact Nat.zero_lt_of_lt hn₀
+    . norm_cast
+      exact lt_add_one n
+
+lemma imo_1985_p6_9_20
+  (f : ℕ → NNReal → ℝ)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (i : NNReal)
+  (hi₁ : f₀ n i = 1 - (↑n)⁻¹):
+   (f n i * (f n i + 1 / ↑n)).toNNReal < (1 - (↑n + 1:ℝ)⁻¹).toNNReal := by
+  rw [hf₁ n i (by linarith), hi₁]
+  refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+  . refine sub_pos.mpr ?_
+    refine inv_lt_one_of_one_lt₀ ?_
+    norm_cast
+    exact Nat.lt_add_right 1 hn₀
+  . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+    rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+    simp
+    refine inv_strictAnti₀ ?_ ?_
+    . norm_cast
+      exact Nat.zero_lt_of_lt hn₀
+    . norm_cast
+      exact lt_add_one n
+
+
+lemma imo_1985_p6_9_21
+  (n : ℕ)
+  (hn₀ : 2 ≤ n):
+   (↑((1:NNReal) - (↑n)⁻¹) * (↑((1:NNReal) - (↑n)⁻¹) + (1:ℝ) / ↑n)).toNNReal < (1 - (↑n + 1:ℝ)⁻¹).toNNReal := by
+  refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+  . refine sub_pos.mpr ?_
+    refine inv_lt_one_of_one_lt₀ ?_
+    norm_cast
+    exact Nat.lt_add_right 1 hn₀
+  . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+    rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+    simp
+    refine inv_strictAnti₀ ?_ ?_
+    . norm_cast
+      exact Nat.zero_lt_of_lt hn₀
+    . norm_cast
+      exact lt_add_one n
+
+
+lemma imo_1985_p6_9_22
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (i j: NNReal)
+  (hi₀ : i = fi n (1 - (↑n)⁻¹))
+  (hj₀ : j = fi (n + 1) (1 - (↑n + 1:NNReal)⁻¹))
+  (hn₂ : f₀ (n + 1) i < f₀ (n + 1) j):
+   fi n (1 - (↑n)⁻¹) < fi (n + 1) (1 - (↑n + 1:NNReal)⁻¹) := by
+  rw [← hi₀, ← hj₀]
+  refine (StrictMono.lt_iff_lt ?_).mp hn₂
+  exact hmo₂ (n + 1) (by linarith)
+
+lemma imo_1985_p6_9_23
+  (n : ℕ)
+  (hn₀ : 2 ≤ n):
+   0 < 1 - (↑n + (1:ℝ))⁻¹ := by
+  refine sub_pos.mpr ?_
+  refine inv_lt_one_of_one_lt₀ ?_
+  norm_cast
+  exact Nat.lt_add_right 1 hn₀
+
+lemma imo_1985_p6_9_24
+  (n : ℕ)
+  (hn₀ : 2 ≤ n):
+  (↑n + 1:ℝ)⁻¹ < (1:ℝ) := by
+  refine inv_lt_one_of_one_lt₀ ?_
+  norm_cast
+  exact Nat.lt_add_right 1 hn₀
+
+lemma imo_1985_p6_9_25
+  (n : ℕ)
+  (hn₀ : 2 ≤ n):
+   ↑((1:NNReal) - (↑n)⁻¹) * (↑((1:NNReal) - (↑n)⁻¹) + 1 / ↑n) < (1:ℝ) - (↑n + (1:ℝ))⁻¹ := by
+  have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+  simp
+  refine inv_strictAnti₀ ?_ ?_
+  . norm_cast
+    exact Nat.zero_lt_of_lt hn₀
+  . norm_cast
+    exact lt_add_one n
+
+lemma imo_1985_p6_9_26
+  (n : ℕ)
+  (hn₀ : 2 ≤ n)
+  (g₀ : (↑n:NNReal)⁻¹ ≤ 1):
+  ↑((1:NNReal) - (↑n)⁻¹) * (↑((1:NNReal) - (↑n)⁻¹) + 1 / ↑n) < (1:ℝ) - (↑n + (1:ℝ))⁻¹ := by
+  rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+  simp
+  refine inv_strictAnti₀ ?_ ?_
+  . norm_cast
+    exact Nat.zero_lt_of_lt hn₀
+  . norm_cast
+    exact lt_add_one n
+
+
+lemma imo_1985_p6_9_27
+  (n : ℕ)
+  (hn₀ : 2 ≤ n):
+   (↑n + 1:ℝ)⁻¹ < (↑n)⁻¹ := by
+  refine inv_strictAnti₀ ?_ ?_
+  . norm_cast
+    exact Nat.zero_lt_of_lt hn₀
+  . norm_cast
+    exact lt_add_one n
+
+lemma imo_1985_p6_10_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₀ : fb = fun (n : ↑sn) => fi (↑n) (1 - 1 / ↑↑n))
+  (hnb₀ : 2 ∈ sn)
+  (nb : ↑sn)
+  (hnb : nb = ⟨2, hnb₀⟩):
+   0 < fb nb := by
+  have g₁: (2:NNReal).IsConjExponent (2:NNReal) := by
+    refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+    . exact one_lt_two
+    . norm_cast
+      simp
+  rw [hfb₀]
+  simp
+  have hnb₁: nb.val = 2 := by simp_all only [one_div]
+  rw [hnb₁]
+  norm_cast
+  rw [NNReal.IsConjExponent.one_sub_inv g₁]
+  let x := fi 2 2⁻¹
+  have hx₀: x = fi 2 2⁻¹ := by rfl
+  have hx₁: f₀ 2 x = 2⁻¹ := by
+    rw [hx₀]
+    have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+    exact g₃ 2⁻¹
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₁)
+
+
+lemma imo_1985_p6_10_2
+  (fi : ℕ → NNReal → NNReal)
+  (sn : Set ℕ)
+  (nb : ↑sn)
+  (hnb₀ : 2 ∈ sn)
+  (hnb : nb = ⟨2, hnb₀⟩):
+   NNReal.IsConjExponent 2 2 := by
+  refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+  . exact one_lt_two
+  . norm_cast
+    simp
+
+
+lemma imo_1985_p6_10_3
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (hnb₀ : 2 ∈ sn)
+  (nb : ↑sn)
+  (hnb : nb = ⟨2, hnb₀⟩)
+  (g₁ : NNReal.IsConjExponent 2 2):
+   0 < fi (↑nb) (1 - (↑↑nb)⁻¹) := by
+  have hnb₁: nb.val = 2 := by simp_all only [one_div]
+  rw [hnb₁]
+  norm_cast
+  rw [NNReal.IsConjExponent.one_sub_inv g₁]
+  let x := fi 2 2⁻¹
+  have hx₀: x = fi 2 2⁻¹ := by rfl
+  have hx₁: f₀ 2 x = 2⁻¹ := by
+    rw [hx₀]
+    have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+    exact g₃ 2⁻¹
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₁)
+
+
+lemma imo_1985_p6_10_4
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (g₁ : NNReal.IsConjExponent 2 2):
+   0 < fi 2 (1 - 2⁻¹) := by
+  rw [NNReal.IsConjExponent.one_sub_inv g₁]
+  let x := fi 2 2⁻¹
+  have hx₀: x = fi 2 2⁻¹ := by rfl
+  have hx₁: f₀ 2 x = 2⁻¹ := by
+    rw [hx₀]
+    have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+    exact g₃ 2⁻¹
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₁)
+
+
+lemma imo_1985_p6_10_5
+  (fi : ℕ → NNReal → NNReal)
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hfb₀ : fb = fun (n : ↑sn) => fi (↑n) (1 - 1 / ↑↑n))
+  (hnb₀ : 2 ∈ sn)
+  (nb : ↑sn)
+  (hnb : nb = ⟨2, hnb₀⟩)
+  (g₂ : 0 < fi 2 (1 - 2⁻¹)):
+   0 < fb nb := by
+  rw [hfb₀]
+  simp
+  have hnb₁: nb.val = 2 := by simp_all only [one_div]
+  rw [hnb₁]
+  norm_cast
+
+
+
+lemma imo_1985_p6_10_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (g₁ : NNReal.IsConjExponent 2 2):
+   0 < fi 2 2⁻¹ := by
+  let x := fi 2 2⁻¹
+  have hx₀: x = fi 2 2⁻¹ := by rfl
+  have hx₁: f₀ 2 x = 2⁻¹ := by
+    rw [hx₀]
+    have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+    exact g₃ 2⁻¹
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₁)
+
+
+lemma imo_1985_p6_10_7
+  (f₀ : ℕ → NNReal → NNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (x : NNReal := fi 2 2⁻¹)
+  (hx₀ : x = fi 2 2⁻¹):
+   f₀ 2 x = 2⁻¹ := by
+  rw [hx₀]
+  have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+  exact g₃ 2⁻¹
+
+
+lemma imo_1985_p6_10_8
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (g₁ : NNReal.IsConjExponent 2 2)
+  (x : NNReal := fi 2 2⁻¹)
+  (hx₀ : x = fi 2 2⁻¹)
+  (hx₁ : f₀ 2 x = 2⁻¹):
+   0 < fi 2 2⁻¹ := by
+  rw [← hx₀]
+  contrapose! hx₁
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₁)
+
+
+lemma imo_1985_p6_10_9
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (g₁ : NNReal.IsConjExponent 2 2)
+  (x : NNReal := fi 2 2⁻¹)
+  (hx₁ : x ≤ 0):
+   f₀ 2 x ≠ 2⁻¹ := by
+  have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₁)
+
+
+
+lemma imo_1985_p6_10_10
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (g₁ : NNReal.IsConjExponent 2 2)
+  (x : NNReal := fi 2 2⁻¹)
+  (hc₁ : x = 0):
+   f₀ 2 x ≠ 2⁻¹ := by
+  have hc₃: f₀ 2 x = 0 := by
+    rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+    norm_cast
+    rw [zero_mul]
+    exact Real.toNNReal_zero
+  rw [hc₃]
+  exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₁)
+
+
+lemma imo_1985_p6_10_11
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (x : NNReal := fi 2 2⁻¹)
+  (hc₁ : x = 0):
+   f₀ 2 x = 0 := by
+  rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+  norm_cast
+  rw [zero_mul]
+  exact Real.toNNReal_zero
+
+
+lemma imo_1985_p6_10_12
+  (sn : Set ℕ)
+  (sb : Set NNReal)
+  (fb : ↑sn → NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (br : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (nb : ↑sn)
+  (g₀ : 0 < fb nb):
+   0 < br := by
+  have g₁: ∃ x, 0 < x ∧ x ∈ sbr := by
+    use (fb nb).toReal
+    constructor
+    . exact g₀
+    . rw [hsbr]
+      simp
+      use fb ↑nb
+      constructor
+      . rw [hsb₀]
+        exact Set.mem_range_self nb
+      . exact congrFun hfr (fb ↑nb)
+  obtain ⟨x, hx₀, hx₁⟩ := g₁
+  have hx₂: br ∈ upperBounds sbr := by
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  exact gt_of_ge_of_gt (hx₂ hx₁) hx₀
+
+
+lemma imo_1985_p6_10_13
+  (sn : Set ℕ)
+  (sb : Set NNReal)
+  (fb : ↑sn → NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (nb : ↑sn)
+  (g₀ : 0 < fb nb):
+   ∃ x, 0 < x ∧ x ∈ sbr := by
+  use (fb nb).toReal
+  constructor
+  . exact g₀
+  . rw [hsbr]
+    simp
+    use fb ↑nb
+    constructor
+    . rw [hsb₀]
+      exact Set.mem_range_self nb
+    . exact congrFun hfr (fb ↑nb)
+
+lemma imo_1985_p6_10_14
+  (sn : Set ℕ)
+  (sb : Set NNReal)
+  (fb : ↑sn → NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (nb : ↑sn)
+  (g₀ : 0 < fb nb):
+   0 < ↑(fb nb) ∧ ↑(fb nb) ∈ sbr := by
+  constructor
+  . exact g₀
+  . rw [hsbr]
+    simp
+    use fb ↑nb
+    constructor
+    . rw [hsb₀]
+      exact Set.mem_range_self nb
+    . exact congrFun hfr (fb ↑nb)
+
+
+lemma imo_1985_p6_10_15
+  (sn : Set ℕ)
+  (sb : Set NNReal)
+  (fb : ↑sn → NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (sbr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (nb : ↑sn):
+   ↑(fb nb) ∈ sbr := by
+  rw [hsbr]
+  simp
+  use fb ↑nb
+  constructor
+  . rw [hsb₀]
+    exact Set.mem_range_self nb
+  . exact congrFun hfr (fb ↑nb)