Non-constant entire function-bounded or not? Show that if $f$ is a non-constant entire function,it cannot satisfy the condition: $$f(z)=f(z+1)=f(z+i)$$
My line of argument so far is based on Liouville's theorem that states that every bounded entire function must be a constant.
So I try-to no avail-to show that if $f$ satisfies the given condition, it must be bounded. I haven't made much progress with this, so any hints or solutions are welcome.

First you can prove that
$$f(z+\mathbb Z(i)) = f(z)$$
so $f$ is determined by it's values on the square $[0, 1+i)$. Entire functions are bounded on precompact sets. QED.
