# Uniform convergence of a functional sequence using Ascoli-Arzela threorem

Assume that $K$ is a compact metric space, real-valued functions $g_n$ defined on $K$ form an equicontinuous family and $g_n(x)$ is convergent for each $x\in K$. It follows by the Ascoli-Arzela theorem that some sequence $g_{k_n}$ of $g_n$ is convergent uniformly.

How to obtain using Ascolli-Arzela theorem that the sequence $(g_n)$ is convergent uniformly?

Your second assumption ($g_n(x)$ converges for every $x$) implies that each (uniformly) convergent subsequence necessarily converges to the pointwise limit $g(x) := \lim_n g_n(x)$.
If $g_n$ would not converge to $g$ the there would be a subsequence which does not, and to that subsequence you can apply Arzela Ascoli again to arrive at a contradiction.