maximum modulus principle implies Liouville's Theorem

Today during the qualifying exam I met this question:

Show that the maximum modulus principle implies the Liouville Theorem.

Well, this is my attempt:

It suffices to show that a bounded entire function can achieve its maximum modulus in complex plane. But I got messed up here. Can anyone give me some ideas?

Solution 1:

Show that the maximum modulus principle implies the Liouville Theorem.

The function

$$f(z) = \frac{z}{1+\lvert z\rvert}$$

which is a homeomorphism between $\mathbb{C}$ and the unit disk shows that the maximum modulus principle alone does not imply Liouville's theorem. We need some more properties of holomorphic functions.

It suffices to show that a bounded entire function can achieve its maximum modulus in complex plane. But I got messed up here. Can anyone give me some ideas?

Something closely related: Riemann's removable singularity theorem.

The Riemann sphere is compact, so every continuous function on the entire sphere ...