An equicontinuous pointwise-bounded family in $C(K,\mathbb R)$ has a finite net in the supremum metric

Choose a finite $\delta$-net $E=\{a_0,\ldots,a_N\}$ in $K$ from the uniform equicontinuity radius for $\varepsilon/3$.

By [L2], every vector $(f(a_0),\ldots,f(a_N))$ lies in one bounded box in $\mathbb R^{N+1}$; cover that box by finitely many coordinate cubes of side less than $\varepsilon/3$.

Choose one member of $\mathcal F$ from each nonempty inverse image of such a cube. Every $f\in\mathcal F$ and its chosen representative differ by less than $\varepsilon/3$ on $E$.

For any $x\in K$, choose $a_i\in E$ with $d(x,a_i)<\delta$ and use equicontinuity for both functions and step 2.1 to obtain $|f(x)-g(x)|<\varepsilon$.

The finitely many representatives form an $\varepsilon$-net, so $\mathcal F$ is totally bounded.