1. The problem statement, all variables and given/known data Given: f is an entire function, Re f(z) ≤ n for all z. Show f is constant. 2. Relevant equations 3. The attempt at a solution So I thought I'd use Liouville's Theorem which states that, if f(z) is entire and there is a constant m such that |f(z)| ≤ m for all z, then f is constant. Looking at the problem I thought to rename Re f(z) as g(z) so then g(z) ≤ n for all z. Then take the modulus of both sides and I get |g(z)| ≤ |n| which is exactly what Liouville's Theorem is asking for. Then g(z) is constant and since I said g(z) = Re f(z), Re f(z) must be constant. I'm not sure how to show from there that f(z) instead of Re f(z) is constant. I'm also not sure my logic followed. I'd appreciate some help, thanks!