Arzelà–Ascoli theorem

Arzelà–Ascoli theorem: Let $K$ be a compact metric space , and let $f_n:K\to\mathbb{R}$ be continuous for all $n$. If the sequence $(f_n)$ is uniformly bounded and equicontinuous , then there exists a subsequence $(f_{n_k})$ and a continuous function $f:K\to\mathbb{R}$ such that $f_{n_k}\to f$ uniformly on $K$.

In the language of the space of continuous functions with the supremum norm , this is a compactness criterion for extracting uniformly convergent subsequences from equicontinuous, uniformly bounded families.