Suppose $\{f_n\}$ is a uniformly bounded sequence of holomorphic functions in $\Omega$ such that $\{f_n(z)\}$ converges for every $z \in \Omega$. Does it necessarily follow that the convergence is uniform on every compact subset of $\Omega$? Here, $\Omega$ denotes a plane open set.
Idea. Perhaps we should apply the dominated convergence theorem to the Cauchy formula for $f_n - f_m$?