# 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.

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Do you know the maximum modulus principle? How might it be adapted to fit this? – GEdgar Dec 12 '11 at 22:21
kahen - I ment sequence, I'll edit the question, thanks. GEdgar - Yes, I know the maximum modulus pronciple, but I don't know how to use it here. – bond Dec 12 '11 at 22:51

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 $h$.
According to the maximum modulus principle the max of $h$ is 0. therefore $h$ is 0 on $|z|=r_n$, meaning that h is 0 everywhere, contradicting the assumption? – bond Dec 13 '11 at 8:39
No. The limit of the maxima in the circles $|z|=r_n$ is $0$. In particular $h$ is bounded... – Julián Aguirre Dec 13 '11 at 14:07