Show that an entire function is a constant

Question

Let $f$ be entire and suppose there is a constant $M>0$ such that $|f(z)|>M$ for all $z \in \mathbb C$. Prove that $f$ is constant.

I think this has something to do with Liouville's theorem but not sure how to go about it!

Answer 1

Show that $g(z):=\frac1{f(z)}$ is constant.

Answer 2

Hint : Apply Liouville theorem to $\frac{1}{f}$.