Why: A holomorphic function with constant magnitude must be constant. How can I prove the following assertion?  
Let $f$ be a holomorphic function such that $|f|$ is a constant.  Then $f$ is constant.  
Edit: The more elementary the proof, the better. I'm working my way through a complex analysis workbook, and by this exercise, the only substantial thing covered has been the Cauchy-Riemann equations. 

Pete's answer is quite sufficient, but I'll add another one which applies to more general maps (namely, complex-valued harmonic). Write $f=u+iv$, where $u$ and $v$ are real harmonic functions. As an exercise with the chain rule, verify the identity
$$\Delta(u^2+v^2)=2|\nabla u|^2+2|\nabla v|^2$$
Conclude that if $u^2+v^2$ is constant, then so are $u$ and $v$.
