91Ó°ÊÓ

Let \(G\) be an open subset of \(\mathbf{C}\). Prove that \(\overline{\mathbf{C}} \backslash G\) is connected if and only if every connected component of \(G\) is simply connected.

Exercise X.19.4. (Vitali's theorem) Let \(\left(f_{n}\right)_{n=1}^{\infty}\) be a locally uniformly bounded sequence of holomorphic functions in a domain \(G\). Assume that the sequence converges at each point of a set which has a limit point in \(G\). Prove that the sequence converges locally uniformly in \(G\).