TITLE: For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.
QUESTION [5 upvotes]: Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset $K\subseteq \Omega$. Then by Cauchy Theorem and Morera Theorem, $f$ is holomorphic. Let $f_n$ and $f'$ be the derivatives of $f_n$ and $f$ respectively.
Prove that $f_n'\to f'$ uniformly on any compact subset $K\subseteq \Omega$. 
How to prove?

REPLY [2 votes]: Not to be taken too seriously! The space $H(\Omega)$ of holomorphic functions on $\Omega$ endowed with the topology of uniform convergence on compact sets is complete and therefore a Frechet space. If you know that the derivative of a holomorphic function is again holomorphic you get from the closed graph theorem that the linear map $D:H(\Omega)\to H(\Omega)$, $f\mapsto f'$ is continuous. That $D$ has closed graph follows e.g. from its continuity as a map $H(\Omega)\to C(\Omega)$.