Period of f given

Question

So the problem states that if $f(z)$ is entire, and satisfies the relation $f(z+i) = f(z)$ and $f(z+1) = f(z)$, show that $f(z)$ is constant. So I was thinking that since any point in $\mathbb{C}$ can be written as $\alpha * 1 + \beta * i $ we can say that $f(z + z_0) = f(z) $ in which case it is constant, but I'm having trouble breaking down the steps, and using the fact that f is entire, which makes me feel like I'm missing something. What should I review to figure this out?

Answer 1

It suffices to consider $f(z)$ for $z \in \{a+bi : a, b, \in [0,1]\}$. Then you can use Liouville's theorem.