Uncountable Poles of a Laplace Transform

Question

I'm trying to prove that a function $F(s)$ of a complex variable resulting from a two-sided Laplace transform can have at most a countable number of poles. I start with a well-known result that a Laplace transform can only give you an analytic (thus holomorphic) function of $s$. Then I reason that if there are uncountably many poles of the Laplace transform, then there must exist some compact subset of the transform's region of convergence that contains infinitely many poles - otherwise, we could have just represented the ROC as a union of a countable number of compact subsets each containing a finite number of poles, which would only give us a countable total number of poles. That's where I run into problems - how do I prove that it's impossible to have a compact set with an infinite number of poles?