Vitali's Convergence Theorem

Theorem
Let $U$ be an open,  connected subset of $\C$.

Let $S \subseteq U$.

Let $\mathcal H \left({U}\right)$ be the space of mappings holomorphic on $U$.

Let $\left \langle{f_n}\right \rangle_{n \mathop \in \N}$ be a normal family of  mappings contained in $\mathcal H \left({U}\right)$.

Suppose that $f_n(z)$ converges to a function $f \in \mathcal H \left({U}\right)$ on $S$ and that $S$ contains a limit point in $U$.

Then the convergence of $f_n$ is uniform on compact subsets of $U$.