Suppose $f(z)$ is entire and $Re(f(z))$ is bounded. Show that $f$ is constant There is a hint that states it might be helpful to consider $exp(f(z))$.  
I don't see why having a real part would imply that the function is constant.  

$\exp(f(z))=\exp(\mathrm{Re\ }f(z))\exp(i\mathrm{Im\ }f(z))$ is bounded Liouville implies that $\exp(f(z))$ is constant and henceforth $f$ is constant.
