Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 4 / 13G / b / Solution

At every regular point of the Riemann sphere, there is a neighborhood containing no poles; at each pole, there is a neighborhood containing no other poles. These neighborhoods cover the sphere. Its compactness gives a finite subcover, each member containing at most one pole. Hence the total number of poles, including a possible pole at infinity, is finite. Equivalently, an infinite pole set would have an accumulation point on the compact sphere, contrary to local regularity or isolation there.