Liouville theorem via the Cauchy estimate

Let $f$ be analytic on the whole complex plane and bounded, say $|f(z)|\le M$ for all $z$. Show that $f$ must be constant.

Show $f'\equiv0$. By Cauchy's integral formula on the circle of radius $R$ about any $z_0$, $|f'(z_0)|\le M/R$. Letting $R\to\infty$ gives $f'(z_0)=0$ for every $z_0$, so $f$ is constant.