Let $p$ be a complex polynomial of degree $n$.

• By analytic continuation arguments applied to any branch of $p^{-1}$, it can be shown that any level curve $\Lambda=\{z:|p(z)|=\epsilon\}$ of $p$ which does not contain a zero of $p'$ is a Jordan curve.

• Since $p'$ has at most $n-1$ zeros, we can find some point $z$ such that the level curve $\Lambda_z$ of $p$ which contains $z$ does not contain a critical point.

• Let $D$ denote the bounded face of $\Lambda_z$. By the maximum modulus theorem applied to $f$ on $D$, $f$ has a zero in $D$. $\Box$