[Math] Equicontinuity implies (pointwise bounded iff uniformly bounded)

Question

Let $X$ be a compact metric space. Suppose that a family $\mathcal{F} \subset C(X)$ is equicontinuous. Prove that $\mathcal{F}$ is uniformly bounded iff it is pointwise bounded.

Can I get a hint?

Answer 1

Of course, the difficult direction is to show that if $\mathcal F$ is pointwise bounded then it is uniformly bounded.

Fix $\delta$ in the definition of equi-continuity corresponding to $\varepsilon=1$. Then cover $X$ by finitely many ball of radius $<\delta$.