Prove Zeros & Poles of Meromorphic Function Have No Limit Point

[Amer]

Prove that the zeros and the poles of a meromorphic function dose not have a limit point

Solution:
Let P be the set of the poles of f suppose that it has a limit point.
p is a pole of f so it is an isolated point such that $$\lim_{z \rightarrow p} \mid f(z) \mid= \infty$$
from isolated point definition there exist an R>0 such that f is analytic at the set B={z : 0<|z-p|<R }
but B is open and p in B, and B intersect U\{p} is not phi so f is not analytic at B
Contradiction

what about the set of zeros, and is my proof right ?

 

[Opalg]

zeroes of analytic functions are isolated | planetmath.org.

 

[Amer]

Thanks for useful link :)