= Solution
Write $C(K)=\bigcup_{n\geq1}\mathcal S_n$ with every $\mathcal S_n$ equicontinuous. Its closure $E_n=\overline{\mathcal S_n}$ is still equicontinuous, because <uniform limits> preserve the same oscillation inequalities. The <Baire category theorem> applied to the <Banach space> $C(K)$ shows that some $E_n$ has nonempty interior. Hence there are $f_0\in C(K)$ and $r>0$ such that the open ball $B(f_0,r)$ lies in $E_n$.
Fix $x\in K$. Equicontinuity of $E_n$ gives a neighborhood $U$ of $x$ on which every member of $E_n$ changes by less than $r/4$. If $y\in U$ were distinct from $x$, the <normality of a compact Hausdorff space> and the <Urysohn lemma> would give $h\in C(K)$ with $h(x)=0$, $h(y)=r/2$, and $\lVert h\rVert_\infty=r/2$. Both $f_0$ and $f_0+h$ belong to $E_n$, yet
$$
|h(y)-h(x)|\leq |(f_0+h)(y)-(f_0+h)(x)|+|f_0(y)-f_0(x)|<r/2,
$$
a contradiction. Thus $U=\{x\}$, so every point is <isolated> and $K$ is a <discrete space>. A compact discrete space is finite, since its cover by singleton open sets has a finite subcover.
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