The Arzela-Ascoli Theorem says that
If $X$ is a compact metric space and $F$ a subset of $C(X)$, then $F$ is compact if and only if $F$ is closed, uniformly bounded, and equicontinuous.
My textbook gives the following corollary:
If $X$ is a compact metric space, and $\langle f_n\rangle_{n\in \mathbb{N}}$ a uniformly bounded, equicontinuous sequence in $C(X)$, then some subsequence converges uniformly on $X$.
Can someone explain how this corollary follows? Obviously the set of all $f_n$ is itself uniformly bounded and equicontinuous, but why must it be closed?
The closure of the set of all $f_n$ is uniformly bounded and equicontinuous, hence compact by Arzela–Ascoli.