An example of an application of Arzelà-Ascoli is that we can use it to prove that the following operator is compact: $ T: C(X) \to C(Y), f \mapsto \int_X f(x) k(x,y)dx$ where $f \in C(X), k \in C(X \times Y)$ and $X,Y$ are compact metric spaces.
To prove that $T$ is compact we can show that $\overline{TB_{\|\cdot\|_\infty} (0, 1)}$ is bounded and equicontinuous so that by Arzelà-Ascoli we get what we want. It's clear to me that if $TB_{\|\cdot\|_\infty} (0,1)$ is bounded then $\overline{TB_{\|\cdot\|_\infty} (0, 1)}$ is bounded too. What is not clear to me is why $\overline{TB_{\|\cdot\|_\infty} (0, 1)}$ is equicontinuous if $TB_{\|\cdot\|_\infty} (0, 1)$ is.
I think about it as follows: $TB_{\|\cdot\|_\infty} (0, 1)$ is dense in $\overline{TB_{\|\cdot\|_\infty} (0, 1)}$ with respect to $\|\cdot\|_\infty$ hence all $f$ in $\overline{TB_{\|\cdot\|_\infty} (0, 1)}$ are continuous (since they are uniform limits of continuous sequences). Since $Y$ is compact they are uniformly continuous. Now I don't know how to argue why I get equicontinuity from this. Thanks for your help.