Show that if $f(z)$ is entire and $\left\vert f(z) \right\vert > M$ for all $z \in \mathbb{C}$, then $f(z)$ is constant.

Question

I am trying to prove that if $f(z)$ is entire and $\left\vert f(z) \right\vert > M$ for all $z \in \mathbb{C}$, then $f(z)$ is constant. Is my proof correct?

Since $f(z)$ is entire, we know that $\frac{1}{f(z)}$ is entire and satisfies $\left\vert \frac{1}{f(z)} \right\vert < 1/M$. So, by Liouville's Theorem, $\frac{1}{f(z)}$, and hence $f(z)$, is constant.

Answer 1

Is my proof correct?

The correct start of the proof should be:

Since $f$ is entire and nowhere zero, we know that $\frac{1}{f}$ is entire ...

Perhaps you meant that without writing it explicitly. You should also distinguish between $f$ (the function) and $f(z)$ (a complex number, the function evaluated at $z$).

Apart from that, the proof is correct.