Show that a non-constant entire function has a dense image.

Question

Let $f$ be a nonconstant entire function and $U$ be an open set in the plane. Show that there is a $z_0$ such that $f\left(z_0\right)\in U$.

This question is an exercise for the Maximum Modulus and Mean Value section. I can't figure out how to prove this. I'm more than sure this requires an application of the mean value theorem, but I don't exactly know how to use it.

Any suggestions/tips on how to proceed?

Answer 1

Hint: consider $1/(f(z) - u)$, and use Liouville's Theorem.