Math 6210 • Real Analysis

Suppose \(\Omega \) and \(\Lambda \) are open subsets of the complex plane. Suppose \(\Omega \) is connected and \(\Lambda \) is a simply connected, proper subset of \(\mathbb C\). If \(\{f_n\}\) is a sequence of holomorphic functions from \(\Omega \) to \(\Lambda \), prove that either \(\{f_n\}\) has a subsequence which converges uniformly on compacta to a holomorphic function on \(\Omega \) or \(\{f_n\}\) itself converges to \(\infty \) at every point of \(\Omega \).