Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 2H / iii

Suppose for contradiction that $f$ has no zero in the closed disc. Thenis a homotopy through loops in $\mathbb{C}\setminus \{0\}$. At $s=1$ it is $\alpha$, while at $s=0$ it is the constant loop with value $f(0)$. A constant loop has winding number zero, so homotopy invariance givescontrary to the hypothesis. Hence