Equicontinuity–boundedness criterion

Equicontinuity–boundedness criterion: Let $K$ be a compact metric space and let $\mathcal{F}$ be a family of functions $f:K\to\mathbb{R}$. Assume $\mathcal{F}$ is an equicontinuous family and is pointwise bounded on $K$. Then $\mathcal{F}$ is uniformly bounded on $K$.

This lemma is a standard compactness-based tool used alongside Arzelà–Ascoli when working in the supremum norm .