Prove: Liouville’s theorem. If f(z) is analytic in the entire complex plane and is bounded (i.e. |f(z)| < M for some constant M) then f(x) must be a constant.

Proof. Using Cauchy’s inequality

with n = 1 and replacing a with z we get

            |f '(z)| M/r

Letting , we observe that |f '(z)| = 0. Thus f(z) = constant.