Normal family theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

This fact is an application of the following pivotal fact/result/idea: Arzela-Ascoli theorem
View other applications of Arzela-Ascoli theorem OR Read a survey article on applying Arzela-Ascoli theorem

Statement

Suppose $U\subset \mathbb {C}$ is an open subset and $f_{\alpha }:U\to \mathbb {C}$ is a family of holomorphic functions such that the $f_{\alpha }$ form a normal family: for every compact subset $K\subset U$, the restrictions of $f_{\alpha }$ to $K$ are uniformly bounded on $K$. Then, there exists a subsequence of the $f_{\alpha }$s, say $f_{n}$, such that $f_{n}$ converge to a holomorphic function, and the convergence is uniform on compact subsets.