Winding-number proof of the fundamental theorem of algebra

For a polynomial $P(z)=a_n{z}^n+\cdots +a_0$ of positive degree, on a sufficiently large circle its leading term dominates the remaining terms. The dominated-perturbation lemma therefore gives winding number $n$ to the loop $P(R{e}^{2\pi it})$. If $P$ had no zero, radial contraction of the input circle would map under $P$ to a homotopy with a constant loop in $\mathbb{C}\setminus \{0\}$, which has winding number zero. This contradiction proves the Fundamental theorem of algebra.