Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 21H / c / Solution

For each $n\ge 0$, defineEvery $f^{(n)}$ is an entire function, hence continuous, so $E_n$ is closed. The hypothesis saysThe complex plane is a complete metric space. The Baire category theorem therefore implies that some $E_N$ has nonempty interior. Thus $f^{(N)}$ vanishes on an open set, and the identity theorem giveson all of $\mathbb{C}$. If $N=0$, then $f=0$; otherwise the power series of $f$ has no term of degree $N$ or higher. Hence $f$ is a polynomial of degree at most $N-1$. This is the pointwise vanishing derivative criterion for a polynomial.