Equicontinuity + pointwise boundedness implies uniform boundedness on compact sets

Let $(K,d)$ be a compact metric space and let $\mathcal{F}\subseteq C(K,\mathbb{R})$ be a family of continuous functions . Assume:

• $\mathcal{F}$ is equicontinuous on $K$, and
• $\mathcal{F}$ is pointwise bounded on $K$ (for each $x\in K$, $\sup_{f\in\mathcal{F}}|f(x)|<\infty$).

Lemma: Then $\mathcal{F}$ is uniformly bounded on $K$; i.e., there exists $M<\infty$ such that $|f(x)|\le M\quad \text{for all } f\in\mathcal{F},\ x\in K.$

This lemma is a standard step in the proof of the Arzelà–Ascoli theorem .