real analysis - Show $a_{n+1}=sqrt{2+a_n},a_1=sqrt2$ is monotone increasing and bounded by $2$; what is the limit?

Let the sequence $\{a_n\}$ be defined as $a_1 = \sqrt 2$ and $a_{n+1} = \sqrt {2+a_n}$.

Show that $a_n \le$ 2 for all $n$, $a_n$ is monotone increasing, and find the limit of $a_n$.

I've been working on this problem all night and every approach I've tried has ended in failure. I keep trying to start by saying that $a_n>2$ and showing a contradiction but have yet to conclude my proof.