= Solution
For a compact space $K$, the <Arzelà-Ascoli theorem> says that a family $\mathcal F\subset C(K)$ is relatively compact in the <supremum norm> if and only if it is pointwise bounded and equicontinuous. In the usual formulation $K$ is compact Hausdorff; the elementary argument below also works for any compact $K$ for real-valued continuous functions.
Pointwise bounded means $\sup_{f\in\mathcal F}|f(x)|<\infty$ for every $x\in K$. Equicontinuity means that for every $x\in K$ and $\varepsilon>0$ there is a neighborhood $U$ of $x$ such that $|f(y)-f(x)|<\varepsilon$ for every $y\in U$ and $f\in\mathcal F$. Relative compactness means compactness of the closure in $C(K)$, where $\|f\|_\infty=\sup_K|f|$. This is a complete metric vector space, since uniform limits of continuous functions are continuous. Equivalently, every sequence in the family has a uniformly convergent subsequence, with limit in $C(K)$, though not necessarily in the original family.
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