Meromorphic functions on the Riemann sphere are rational

Let \(f\) be meromorphic on \(\CC\) such that \(\infty\) is either a removable singularity or a pole of \(f\), in the sense of singularities at infinity. Then \(f\) is a rational function: \(f=P/Q\) for polynomials \(P,Q\in\CC[z]\) with \(Q\neq0\).