Limit of a sequence of fixed points also a fixed point?

Question

Suppose I have a continuous function $f : [0,1]^n \rightarrow [0,1]^n$ (maybe $n$ is infinite). Suppose I have a sequence $\{a_n\}_{n=1}^\infty$ of points in $[0,1]^n$ where each $a_n$ is a fixed point of $f$, and that $\{a_n\}$ converges to a value $a$. Is $a$ also a fixed point of $f$?

Thank you!

Answer 1

$$f(a)=f\left(\lim\limits_{n\to\infty}a_n\right)=\lim\limits_{n\to\infty} f(a_n)=\lim\limits_{n\to\infty} a_n=a$$

Answer 2

Consider the function $g:[0,1]^n\to\mathbb{R}^n$, $g(x)=f(x)-x$. It is continuous, $g(a_n)=f(a_n)-a_n=0$ for all $n\in\mathbb{N}$, therefore $0=\lim_{n\to\infty} g(a_n)= \lim_{n\to\infty}f(a_n) - \lim_{n\to\infty}a_n = f(a)-a$, i.e. $a$ is also a fixed point of $f$.