= Solution
The appropriate <homotopy invariance of winding number> is: if $H:[0,1]^2\to\mathbb C\setminus\{0\}$ is continuous and $H(s,0)=H(s,1)$ for each $s$, then all loops $t\mapsto H(s,t)$ have the same <winding number>. The basepoint may move; every intermediate curve must close and avoid zero.
<Compactness> gives $m=\min_{s,t}|H(s,t)|>0$. <Uniform continuity> gives $\delta>0$ such that $|s-s'|<\delta$ implies $|H(s,t)-H(s',t)|<m$ for every $t$. Part (d) then makes the <winding numbers> equal for nearby $s,s'$. Subdivide $[0,1]$ into steps smaller than $\delta$ and chain these equalities. Therefore
$$
\boxed{w(H(0,\cdot))=w(H(1,\cdot)).}
$$
This proves the theorem directly for continuous loops, without requiring an integral formula or smooth <homotopy>.
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