Uniform limit of holomorphic functions is holomorphic

Statement

Suppose $f_{n}$ is a sequence of holomorphic functions on an open subset $U \subset C$ , and $f_{n} \to f$ pointwise, with the convergence being uniform on compact subsets. Then $f : U \to C$ is a holomorphic function, and moreover, for any $k$ , we have:

$f_{n}^{(k)} \to f^{(k)}$

with this convergence also being uniform.

Statement

Facts used