Derivative of Sequence of Holomorphic Functions

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

Let $\left \langle{f_n}\right \rangle$ be a sequence of holomorphic mappings $f_n: U \to \C$.

Let $\left \langle{f_n}\right \rangle$ converge pointwise to some function $f: U \to \C$.

Let $\left \langle{f_n}\right \rangle$ converge uniformly on  compact subsets of $U$.

Then $f$ is holomorphic on $U$.

Further, the sequence of derivatives $\left \langle{f_n'}\right \rangle$ converges to $f'$ on $U$.

This convergence is uniform on compact subsets of $U$.