We have proved Arzela-Ascoli Theorem as a family of functions in $C(X)$ where $X$ is a compact metric space is relative compact iff it is equibounded and equicontinuous.
We’ve given a corollary for this. Every equicontinuous and equibounded sequence of $C(X)$ has an uniformly convergent subsequence.
I’ve seen the Rudin’s proof but there is some details like first countablity. Is there any basic and short way to obtain this corollary from the theorem we proved?
I’ve seen some proofs more and I’ve tried to do it but I’m not able to. If someone illuminate me on this shortly, I appreciate.
Thanks in advance