Let $f(z)$ be an entire function with $|f(z)|\leq {\frac{1}{|{\rm Im}(z)|}}$ for all $z$. Show that $f\equiv 0$

Question

I've tried to use $e^{f(z)}$ like the question here: $f$ is an entire function with ${\rm Im}\ f\geq 0$.

However, I'm stuck here. I've learned Liouville's theorem; it's sufficient to show that $f$ is bounded, and it's not difficult to show that the constant is $0$, but how do I show $f$ is bounded?