Let $U\subseteq \mathbb{C}$ be an open set, and let $f_n:U\to \mathbb{C}$ be a family of continuous functions on $U$. If $f_n$ converges to a function $f$ locally uniformly on $U$, then the limit $f$ is ==1;;also continuous. ==

PROOF:

1. Use that uniform limits of continuous functions are continuous
2. Apply glueing lemma