calculus - How to find this limit and prove it rigorously: $sqrt{2 + sqrt{2+sqrt{2+sqrt2...)}}}$?

$\sqrt{2 + \sqrt{2+\sqrt{2+\sqrt2...)}}}$.

Pretty classic question, I think - and the limit is equal to 2.

But how do I prove this rigorously? An epsilon-delta proof wouldn't work, since I wouldn't know the limit is equal to 2 - the question asks, if the limit exists, compute it. This was for an old analysis exam, not a calculus class, so I feel that I can't just set the above = some number L, and then make algebraic manipulations on both sides of the equation, until I get what I want. We can't assume the limit exists, I think.

Thanks,