 Solution

In the supremum-norm metric, $S$ is totally bounded if for every $\epsilon >0$ finitely many norm balls of radius $\epsilon$ cover it. It is uniformly bounded if some $M$ satisfies $|f(x)|\le M$ for all $f\in S,x\in K$. It is equicontinuous if for each $x\in K$ and $\epsilon >0$ there is a neighbourhood $U$ of $x$ with $|f(y)-f(x)|<\epsilon$ for all $f\in S,y\in U$. This definition works for compact Hausdorff $K$ without assuming it is metrizable.

Suppose $S$ is totally bounded. A finite radius-one net makes $\Vert f{\Vert }_{\infty }$ uniformly bounded by one plus the largest norm of a net centre. For equicontinuity, choose a finite $\epsilon /3$ net. Continuity of each centre gives a common neighbourhood of $x$ on which all centre values vary by less than $\epsilon /3$. Approximating any $f$ by its centre and using the triangle inequality at $x$ and $y$ bounds its variation by $\epsilon$. Thus both conditions follow.

Conversely suppose those two conditions hold. For each $x$, choose a neighbourhood $U_x$ on which every function varies from its value at $x$ by less than $\epsilon /3$. Compactness provides a finite cover $U_{x_1},\dots ,U_{x_m}$. The vectors $(f(x_1),\dots ,f(x_m))$ form a bounded subset of ${\mathbb{C}}^m$, so partition their possible values into finitely many boxes of diameter less than $\epsilon /3$ in the maximum-coordinate norm. From each occupied box choose one function as a representative. If $f$ and its representative $h$ have sample values in the same box, and $y\in U_{x_j}$, thenThese finitely many representatives are a norm net, proving total boundedness is equivalent to uniform boundedness and equicontinuity. This supplies the finite-net part of the Arzelà-Ascoli theorem directly.

For a bounded set that is not totally bounded, take the standard unit vectors $S=\{e_n:n\ge 1\}$ in the Banach space ${\ell }^2$. Their norms are one and their pairwise distances are $\sqrt{2}$. A ball of radius less than $\sqrt{2}/2$ contains at most one of them, so finitely many such balls cannot cover $S$.