Liouville's theorem question - show f is constant

Question

Assume that $f: \mathbb C \to \mathbb C$ is entire such that $\left|f\left(z_{0}\right)\right|\neq1$ for all $z\in\mathbb C$. Show that f is constant.

I know either $\left|f\left(z\right)\right|<1$ for all $z\in\mathbb C$ or $\left|f\left(z\right)\right|>1$ for all $z\in\mathbb C$ using Intermediate value theorem.

Afterwards, I should be able to apply Liouville Thm. Hence, if $\left|f\left(z\right)\right|<1$ for all $z\in\mathbb C$ than f is constant (by Liouville Thm) but I don't know what to do with the other option.

Thanks!

Answer 1

In the case of the other option, apply Liouville's theorem to $\dfrac1f$.