Can it be proved that a Meromorphic function only has a countable number of poles?

Solution 1:

Ad 1. Yes, it is part of the definition or a consequence of the definition that a pole is an isolated singularity of the function.

Ad 2. Yes, since poles are isolated singularities, a meromorphic function can have at most countably many poles. That is not hard to prove: Every compact subset $K$ of the domain of the function can only contain finitely many poles [otherwise the set of poles would have a limit point in $K$], and every open subset of $\mathbb{C}$ (or $\widehat{\mathbb{C}}$) is the union of countably many compact subsets (since it is locally compact and second countable), so the set of poles is a countable union of finite sets.

Solution 2:

Poles are by definition isolated singularities, so there is an open disk of positive radius around each of them in which no other singularity occurs. This is sufficient to prove there are at most countably many poles in the complex plane (or in some open domain in the complex plane). This can be seen as a corollary to no uncountable sum of positive real numbers (the areas of these disks) can be finite.