Prove any sequence a set has a uniformly convergent subsequence using Arzela-Ascoli Theorem

Question

Consider the subset of $C([0, 1])$ given by $$S = \{f \in C([0, 1]) : f(0) = 0, \int_{0}^{1} |f'(x)|^2 dx \leq 1 \}$$ How to prove that any sequence in this set $S$ has a uniformly convergent subsequence using Arzela-Ascoli Theorem (https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem). I guess one can show that $S$ is uniformly bounded and equicontinuous, but not sure how.

Answers very much appreciated, thanks.

Answer 1

@Nate Eldredge already said it all: \begin{align*} |f(z)-f(y)|&=\left|\int_{y}^{z}f'(x)dx\right|\\ &\leq\int_{y}^{z}|f'(x)|dx\\ &=\int_{0}^{1}|f'(x)|\chi_{[y,z]}dx\\ &\leq\left(\int_{0}^{1}|f'(x)| ^{2}dx\right)^{1/2}|z-y|^{1/2}, \end{align*} for $z\geq y$, the rest should be straight forward.