(a) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map such that $\gamma(0)=\gamma(1)$. Define the winding number $w(\gamma ; 0)$ of $\gamma$ about the origin. State precisely a theorem about homotopy invariance of the winding number.

(b) Let $B=\{z \in \mathbb{C}:|z| \leqslant 1\}$ and let $f: B \rightarrow \mathbb{C}$ be a continuous map satisfying

$$|f(z)-z| \leqslant 1$$

for each $z \in \partial B$.

(i) For $0 \leqslant t \leqslant 1$, let $\gamma(t)=f\left(e^{2 \pi i t}\right)$. If $\gamma(t) \neq 0$ for each $t \in[0,1]$, prove that $w(\gamma ; 0)=1$.

[Hint: Consider a suitable homotopy between $\gamma$ and the map $\gamma_{1}(t)=e^{2 \pi i t}$, $0 \leqslant t \leqslant 1 .]$

(ii) Deduce that $f(z)=0$ for some $z \in B$.