Prove that if $f$ is an entire function with $\lim_{z \to \infty}|f(z)| = \infty$, then $f$ has at least one zero.

Question

Prove that if $f$ is an entire function with $\lim_{z \to \infty}|f(z)| = \infty$, then $f$ has at least one zero.

I was trying to apply the limits involving point of infinity and putting $z$ to $1/z$. However, I could not proceed. Kindly help in my assignment.

Answer 1

If $f$ has no zeros then $\frac 1f$ is an entire function. Apply Liouville's Theorem to this function.