 Solution

Brouwer's fixed-point theorem in the plane states that every continuous map from a closed disk to itself has a fixed point. Equivalently, the same holds for every nonempty compact convex subset of ${\mathbb{R}}^2$.

On the closed unit disk defineFor $|z|\le 1$,so $F$ maps the disk continuously into itself. Brouwer supplies $z$ with $F(z)=z$, which is equivalent toThus this polynomial has a root with $|z|\le 1$.