Montel's Theorem

Theorem
Let $U \subseteq \C$ be an open subset of the complex numbers.

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

Then a family of mappings $\mathcal F \subseteq \mathcal H \left({U}\right)$ is normal $\mathcal F$ is locally bounded.