Is an Entire Function Satisfying f(z+i)=f(z) and f(z+1)=f(z) Constant?

[playa007]

Homework Statement

if an entire function satisfies f(z+i)=f(z) and f(z+1)=f(z), must the function be constant?

Homework Equations

The Attempt at a Solution

It's true that f(0) = f(k) = f(ik) where k is an integer. I'm wondering whether I can apply Liouville's theorem into this somehow or if it's not constant at all on how to construct a counterexample.

 

[eok20]

Hmm...well I think you can show that all of the function values are determined by the function values on a certain compact set. Since the continuous image of a compact set is bounded, the function must be bounded and so you can apply Liouville's theorem.