Prove that this entire function $f$ is constant

Question

I need help for this problem:

If $f:\mathbb{C}\to\mathbb{C}$ is entire function such that $f(z+1)=f(z)$ and $|f(z)|<e^{|z|}$, $\forall z\in\mathbb{C}$, prove that $f$ is constant function.

[Hint: Look function $g(z)=\frac{f(z)-f(0)}{\sin \pi z}$]