In order to work with Bergman Spaces, I am trying to understand the next assertion
Let $G$ be a complex domain. If $(f_n:G \to \mathbb{C})_n$ is a sequence of analytic functions that is uniformly Cauchy on any closed disk in $G$, then there exist an analytic function $g$ on $G$, such that $f_n(z)\to g(z)$ uniformly on compact subsets of $G$.
The text that I am reading says that the assertion follows by the classical work on analytic functions, however I do not remember any theorem who states something even similar.
My question is:
Is there a complex analysis thm that states exactly this? If yes, which one? If no, how can I prove it with the classical complex analysis technics (like Cauchy's formula, Morera's Thm, etc)?
I think the argument goes as follows: Fix any compact subset $K$ of $G$. View $\{f_n\}$ as a sequence of continuous function on $K$, and therefore a sequence in $\mathcal{C}(K)$. $K$ being compact implies that $\mathcal{C}(K)$ is complete. Then by the hypothesis, the Cauchy sequence $\{f_n\}$ must converge uniformly to a function $g$ in $\mathcal{C}(K)$. Now use Morera's theorem to conclude that $g$ is actually also a holomorphic function on $G$.