I remember doing this exercise once and using a Cantor's diagonal argument somewhere, but I can't remember why I did all that. Take a look at this and let me know if something's off (It's pretty simple, which makes me wary):

Pick with . There exists an such that for all (the closed punctured disk or radius and center ), and by Hurwiz theorem there exists such that for all the functions have the same zeroes as in the disk, so there exists with . It now suffices to prove that , but this follows since is compact (therefore sequentially compact) and the fact that the sequence can only have one accumulation point by how we picked our disk and the fact that whenever and .

Check it carefully, maybe I overlooked something.