Let $U\subseteq \mathbb{C}$ be an open subset, and $f_n:U\to \mathbb{C}$ be holomorphic. Then if $f_n\to f$ locally uniformly, then the limit function $f$ is 1;;holomorphic. Moreover, the higher derivatives of $f_n$ converge locally uniformly

PROOF:

1. Use previous lemma to conclude $f$ is continuous.
2. Apply Morera theorem and Cauchy's theorem to show that $f$ is holomorphic.
3. Use the theorem of Goursat to argue the higher derivatives also converge to a holomorphic function, and this holomorphic function is the respective derivative of $f$.