# Convergence of a sequence holomorphic functions

Let $f_n\in\mathcal{O}(\Omega)$ be a sequence of holomorphic functions, s. th. $f_n\rightarrow f$ pointwise in $\Omega$. Show that exist open and dense set $\Omega'\subset \Omega$ such that $f_n$ is locally uniformly bounded in $\Omega'$.

I should use Baire theorem, but I don't have any idea how construct required family of (open and dense) sets.

## 1 Answer

This is Osgood's theorem, proved in this 1901 paper. His proof goes as follows, slightly paraphrased and modernized:

Let $\Omega'$ be the set of points $z$ such that $f_n$ is locally bounded in a neighborhood of $z$. By definition $\Omega'$ is open, so assume that it is not dense. Then there exists an open disk $D \subset \Omega \setminus \Omega'$. Let $P_m$ be the set of points $z \in D$ for which $|f_n(z)| \le m$ for all $n$. Since $f_n \to f$ pointwise, we know that $D = \bigcup_{m=1}^\infty P_m$. Any interior point of any $P_m$ is a point in $\Omega'$, so by assumption there are no interior points for any of the sets $P_m$. Since each $P_m$ is closed by definition, this implies that each $P_m$ is nowhere dense in $D$, contradicting Baire's theorem.