A cauchy sequence question

Question

Let $a_1=\sqrt 2$ and let $n\ge 2$ be defined recursively by the formula

$a_{n+1}=\sqrt{2+\sqrt{a_n}}$

How can I prove that $a_n$ is a cauchy sequence and conclude that $a_n$ converges?

Answer 1

One possible path: You may show that the function $f(x)=\sqrt{2+\sqrt{x}}$ is a Lipschitz contraction of $I=[\sqrt{2},2]$, i.e. $f(I)\subset I$ and $\forall x,y\in I: |f(x)-f(y)|\leq L|x-y|$ for some $L<1$ (one may here take e.g. $L=1/8$).

Answer 2

Hint: prove that the sequence is monotonically increasing and bounded.