An entire function with bounded $n$th derivative is a polynomial of degree at most $n$

Suppose $f$ is entire and $f^{(n)}$ is bounded on $\CC$. Show that $f$ is a polynomial of degree at most $n$.

Solution

By Liouville or MMP, $f^{(n)}$ is bounded and entire and thus constant. Integrating a constant $n$ times yields a polynomial of degree t most $n$.