I have been thinking about the following exercise from some complex analysis lecture notes I found online for a while but I can't seem to be able to conclude the desired result. The exercise says that
If $A$ is an open subset of the complex plane, and if $g: A \rightarrow \mathbb{C}$ is analytic, then there is a sequence of rational functions $(f_n)$ such that no pole of the $f_n$'s lies in $A$ (that is, that the poles are in $\overline{\mathbb{C}} \setminus A$) and such that $f_n \rightarrow g$ uniformly on compact subsets of $A$.
I know that I have to apply Runge's theorem here, but the problem is that Runge's theorem tells me that the sequence of functions $f_n$ converges uniformly to $g$ on a compact set $K \subseteq A$ and not on the whole $A$.
I thought that maybe I can approximate the open set $A$ by compact sets $K_n$ and take elements from a sequence $f_{n,k}$ that converges uniformly on each of the compacts, and then any given compact $K\subset A$ would have to be contained in one of the compacts $K_n$ approximating $A$ but I'm not sure if this is possible of if it this works.
I would appreciate any help with this exercise. Thanks.