Derivative of Sequence of Holomorphic Functions

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

Let $\sequence {f_n}$ be a sequence of holomorphic functions $f_n: U \to \C$.

Let $\sequence {f_n}$ converge pointwise to some function $f: U \to \C$.

Let $\sequence {f_n}$ converge uniformly on compact subsets of $U$.

Then $f$ is holomorphic on $U$.

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

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