Is $C(X,S)$ $\sigma$-compact ($X$ compact, $S$ is a complete metric space)?

Question

Suppose $X$ is a compact metric space and $S$ is a complete separable metric space. Let $\mathcal{C}(X,S)$ be the space of all continuous functions from $X$ into $S$. Prove that $\mathcal{C}(X,S)$ is $\sigma$-compact. (I read this statement on Kallenberg's book Foundations of Modern Probability, where it is stated as a consequence of Arzela-Ascoli's theorem. A proof of this escapes me.)

Answer 1

This is false: Suppose $C([0,1],\mathbb R)= \cup K_n, n \in \mathbb N,$ where each $K_n$ is compact in $C([0,1],\mathbb R).$ Now $C([0,1],\mathbb R)$ is a complete metric space, so by Baire, some $K_n$ has nonempty interior. But any open ball in $C([0,1],\mathbb R)$ contains a countable set of functions whose distances from each other are greater than some fixed positive number. That violates compactness.