I was told to find $x$ if I know ($a_n$)'s limit while $a_1=x$ and $a_{n+1}=x^a_n$, specifically for $\lim_{n\to\infty}{a_n}=2$ and then for different limits. It is pretty clear that $x=\sqrt{2}$, but I can't for the world of me think of a way to prove it (by the way, we were never taught a statement saying that if $\lim_{n\to\infty}{a_n} = L$ then $\lim_{n\to\infty}{x^{a_n}} = x^L$, I'm not sure if it's true, but if it is I think I have to prove it).
Any help will be greatly appreciated! Thanks.