Easy examples of the Arzela-Ascoli Theorem

Question

Let $X$ be a compact metric space. $M \subseteq C(X)$ is relatively compact if and only if $M$ (i.e. its elements) is equicontinuous and uniformly bounded.

I've been told that this theorem gives me a characterization of the relatively compact subsets of $M$. Could somebody provide some (easy) examples for this? I do understand the implication, but I fail to see where I can use this in an easy and obvious way. I tried Wikipedia, but those examples seem complicated.

Any help is appreciated.

Answer 1

Try $X = [0,1]$, $M$ the set of differentiable functions $f$ on $X$ such that $|f(x)| \le 1$ and $|f'|(x) \le 1$ everywhere.