I am doing problem 3 from section 45 in Munkres. The problem is Prove Arzela's Theorem, which states: Let $X$ be compact: let $f_n \in \mathcal{C}(X,\mathbb{R}^k)$. If the collection $\{f_n\}$ is pointwise bounded and equicontinuous, then the sequence $f_n$ has a uniformly convergent subsequence. Here is a sketch of my proof:
Let X is compact and $\{f_n\}\subseteq \mathcal{C}(X,\mathbb{R}^k)$. Since $\{f_n\}$ is pointwise bounded and equicontinuous by Ascoli's Theorem $\overline{\{f_n\}}$ is compact. Since $\overline{\{f_n\}}$ is a compact subset of a complete metric space it's complete. Since $\overline{\{f_n\}}$ is compact then the sequence $\{f_n\}$ has a convergent subsequence $\{f_{n_i}\} \to f$. Since we are in the uniform metric, this subsequence converges uniformly.
$\endgroup$ 81 Answer
$\begingroup$The argument should say: as we have a compact subset of a metric space, the subset is sequentially compact and so we have a convergent subsequence for the sequence $(f_n)_n$ (completeness is irrelevant as we don't have a Cauchy sequence). The rest seems correct.
$\endgroup$
