prove that if $|f(z)|\geq |z|+|\sin(z)|$ then it cannot be an entire function

by hash man   Last Updated August 01, 2020 14:20 PM

Problem: Prove that if $\forall z \in \mathbb{C}.|f(z)|\geq |z|+|\sin(z)|$ then it cannot be an entire function.

I thought about claiming that $f$ must be a polynomial because it has a pole in infinity, but I stuck why it polynomial cannot satisfy this property.