Q:

Limit of a sequence $\sqrt{1+\sqrt{\underbrace{1+ \ldots +\sqrt{1}}_{n}}}$

$\sqrt{1+\sqrt{\underbrace{1+ \ldots +\sqrt{1}}_{n}}}$
I think the limit is $2$ for intuitive reasons.
Now, to prove it, I need to use the definition of a sequence's limit.
So, I need to solve the following inequality
$|\sqrt{1+\sqrt{\underbrace{1+ \ldots +\sqrt{1}}_{n}}}-2| < \epsilon$
for an arbitrary $\epsilon$.
But I'm not sure it is possible.
One implication of the inequality is 
$\sqrt{1+\sqrt{\underbrace{1+ \ldots +\sqrt{1}}_{n}}} < \epsilon + 2$
What I found out is the same as
$ \underbrace{( \ldots ((}_{n-1}\epsilon + 2\underbrace{)^2 - 1)^2 \ldots )^2}_{n-1} - 2 > 0$
But I'm not sure it is solvable in this form.

A:

Hints: Suppose the limit exists, and denote it by $L$. Then notice that $L=\sqrt {1+L}$, from which you get $L^2-L-1=0$ and you can find the roots to obtain candidates for the limit. If you can then argue why the limit does in fact exist, and you can eliminate one of the limits, then you will have found the answer. In any case, $2$ does not solve the equation, so the limit is certainly not $L=2$. (There are better ways of showing that a sequence converges than playing with $\varepsilon $. Ask yourself, is the sequence monotone? is it bounded?) 

