Let $\Gamma$ be a bounded region in $\mathbb C$ and let $\Sigma$ be a closed set contained in $\Gamma$. Suppose $f$ is a function that is holomorphic on $\Gamma$ apart from singularities where it has poles. Show that the number of singularities within $\Sigma$ is finite.
I know that the singularities are isolated point but I can't see the finiteness of them;
is there a charitable soul who can help me?
Assume for contradiction that there infintely many poles within $\Sigma$. Then we could make a convergent sequence of points within $\Sigma$ which were singularities of the function (why?). But then, $f$ would not be meromorphic since the limit $z_0$ of this sequence would be a singularity, but not an isolated one since every neighborhood of $z_0$ would contain infinitely many singularities.