Locally uniform limits of holomorphic functions are holomorphic

Let \(\Omega\subseteq\CC\) be open, and let \((f_n)_{n\in\NN}\) be holomorphic functions on \(\Omega\) that converge to \(f\) uniformly on every compact subset of \(\Omega\). Then \(f\) is holomorphic on \(\Omega\), and \(f_n'\to f'\) uniformly on every compact subset of \(\Omega\).