[Math] Equicontinuity and $L^2$ convergence imply uniform convergence

Question

I'm currently working through an old Paper of Garsia, Rodemich and Rumsey (A Real Variable Lemma) and theres one thing i don't get. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of continuous real valued functions on $[0,1]$ and the sequence of partial sums $S_m(t)=\sum_{n=1}^m f_n(t)$ converges in $\mathrm{L}^2([0,1])$ and is equicontinuous. Does this imply the uniform convergence of $(S_m)_{m\in\mathbb{N}}$ in $[0,1]$?

Answer 1

Since $S_n(t)$ converges in $L^2$ to a function $S(t)$, it is bounded at least at a point. Since it is equicontinuous, every subsequence, by Ascoli-Arzelà, has a sub-subsequence that converges uniformly. The limit is the same function $S(t)$, hence $S_n$ itself converges uniformly.