 Meromorphic functions on the sphere are rational

Poles are isolated, including in the coordinate $w=1/z$ around infinity. Compactness of the Riemann sphere therefore leaves only finitely many poles. Subtract the finite principal parts at finite poles and the polynomial principal part at infinity. The remainder is entire with a finite limit at infinity, hence bounded and constant by Liouville's theorem. The original function is a sum of a polynomial and finitely many partial fractions, so is rational.