A periodic entire function which must have a fixed point

Question

I would like to check my work on the following problem:

Suppose $f(z)$ is a non-constant periodic entire function satisfying $f(z+1)=f(z)$. Show that $f(z)$ has a fixed point.

So my attempt is: Suppose $f(z)$ does not have a fixed point. Then $g(z)=f(z)-z$ is entire and never $0$. We compute $g(z+1)=f(z+1)-(z+1)=f(z)-z-1=g(z)-1$. But, since $g(z+1)\neq 0$ then $g(z)\neq 1$ for all $z$, hence $g(z)$ is a nonconstant entire function omitting $2$ values, contradicting Picard's theorem.

Looks good?

Answer 1

Looks solid! Sadly, I'm only seeing it well after the OP, but to cut down on unanswered questions, here we go!