Let $X$ be a compact metric space. Prove that an equicontinuous subset of $C(X)$ is uniformly bounded if it is pointwise bounded.
I'm not looking for the answer. I know all the relevant definitions but I'd like a hint as to what $X$ being compact implies.
We'll denote members of the equicontinuous family by $f_\alpha$.
Let $M_x >0$ satisfy $|f_\alpha(x)|< M_x$ for all $\alpha$ (which exists because of the pointwise bound).
Now fix $x$ in $X$. Equicontinuity implies there is some $\delta$ such that $d(x,y)< \delta \implies |f_\alpha(y)| < M_{x} + 1$.
Consider the set $B_x = B(x,\delta)$. It is (and here's the hint) an open set containing $x$.