Locally uniform limit theorem for holomorphic functions

Statement

Let $f_n$ be holomorphic on a common open set $\Omega\subseteq\mathbb C$ . If $f_n\to f$ uniformly on every compact subset, then $f$ is holomorphic, and every fixed derivative $f_n^{(k)}\to f^{(k)}$ uniformly on compact subsets.

Proof through a surrounding circle

Choose a closed disc inside $\Omega$ . Uniform convergence on its boundary passes the limit through the Cauchy integral formula. On a smaller disc the kernel and all its fixed-order derivatives are uniformly bounded, so the resulting integral is holomorphic and can be differentiated. A finite collection of smaller discs covers any given compact subset. A common domain and local uniform convergence are essential; pointwise convergence alone is insufficient.

References

Dan Romik, Complex Analysis, §7.