Prove that exist $z_0 \in \mathbb C$ that satisfy $f(z_0)=0$. I would be glad to get some help with this question:
Let $f(z)$ be an entire function. Assume that there exists a
monotonous increasing and unbounded sequence  $\{r_n\}$ such that $\lim\limits_{n \to \infty} \min\limits_{|z|=r_n} |f(z)|=\infty$.  I want to show that there exists a $z_0 \in \mathbb C$ that satisfies $f(z_0)=0$.
I'd especially like to know how to use that fact about the sequence.
Thanks.

Assume $f(z)\ne0$ for all $z\in\mathbb{C}$. Then $h(z)=1/f(z)$ is also an entire function. Apply the maximum modulus principle to $h$.
