Must a converging sequence converge to a fixed point?

Question

I am trying to show that the infinite tetration $\lim_{n \to \infty}$$^{n}x$ converges for real values of $x$ within $[e^{-e},e^{\frac{1}{e}}]$. To do so, I examined the fixed points of the following recursive function. $$^{n+1}x=x^{^{n}x}$$ Then, the desired result is obtained by examining the value of the iteration function's derivatives. However, I wonder if the following proposition is true: if a sequence $f$ converges, then it must converge to a fixed point. If it is true, how shall I prove it? Thank you!

Answer 1

If the requirements of the Banach fixed-point theorem are fulfilled this is true. If you have a sequence of a function being iterated and the function is constricting and invariant then the limit of the sequence will be that function’s fixed point.

Answer 2

If the function is continuous, what you claim is true. You can simply observe that if $x_n \to L$, then, taking limits on $x_{n+1} = g(x_n)$, leads to $L = g(L)$.