Suppose $f(z)$ is an entire function in $C$, and $|f(z)| ≥ M$, $∀z ∈ C$ for a positive constant $M$. Prove that $f(z)$ must be a constant function.
Hint: Liouville’s theorem may be applied in a novel way.
I am assuming that this question is basically asking to prove Liouville's theorem because that is exactly what the theorem says.
I copied this from Wikipedia to make sure I am not forgetting anything or looking at a different theorem.
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number $M$ such that $|f(z)| <= M$ for all $z \in {C}$ is constant. Equivalently, non-constant holomorphic functions on $C$ have dense images.