Proof of Arzela's Theorem
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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.
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