Locally uniformly bounded holomorphic families are equicontinuous and normal

Let \(\Omega\subseteq\CC\) be open and let \(\mcf\) be a family of holomorphic functions on \(\Omega\) that is uniformly bounded on compact subsets of $\Omega$. Then \(\mcf\) is equicontinuous on every compact subset of \(\Omega\), and \(\mcf\) is a normal family.