Show that the sequence $\sqrt{2}, \ \sqrt{2+\sqrt{2}}, \ \sqrt{2+\sqrt{2+\sqrt{2+}}}...$ converges and find its limit. [duplicate]

I'm not sure why solving $f'(x)=0$ should be significant.

You can instead think like this. Let's do a little exploring.

If $a_n$ converges to a limit $a$, then $f(a_n)$ should converge to $f(a)$ (by continuity of $f$).

But then $f(a_n)=a_{n+1}$ should also converge to $a$.

So $f(a)=a$, which you can solve, for a solution that is $> a_1$.

Having the actual solution for $a$ in your hands, you can then attempt to prove by induction that $a_n \le a$ for all $n$.

From that you conclude that $a_n$ converges. And from that, together with continuity of $f$ and uniqueness of the solution $> a_1$, you can conclude that $a_n$ converges to $a$.