<s>[INST] Proving $\sqrt{a+\sqrt{a+\sqrt{a+\sqrt{a.....}}}}-\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a....}}}}=1$ I checked many values of this inequality, but I don't have the complete proving.
$$\sqrt{a+\sqrt{a+\sqrt{a+\sqrt{a.....}}}}-\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a....}}}}=1$$ if $$a>1$$
 [/INST] Define $x = \sqrt{a + \sqrt{a + \sqrt{a +\cdots}}}$ and $y = \sqrt{a - \sqrt{a - \sqrt{a -\cdots}}}.$ Then we have $x^2 = a+ x$ and $y^2 = a-y$.
Now compute $(x+y)(x-y) = x^2 - y^2 = x+y$. Cancelling $x+y$ from both sides gives $x-y = 1$, which is the result you're after.
(Of course, I've ignored convergence issues here and just worked formally).
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