User:Caliburn/s/6/Proof 1

Proof
Note that for each $z \in \C$, we have:


 * $\size {\map \Re {\map f z} } = \size C < 2 \size C$

So:


 * the real part of $f$ is bounded.

From Entire Function with Bounded Real Part is Constant, we have:


 * $f$ is constant.