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 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_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)) (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) : 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 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_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 : 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) (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)