Solution
= Solution
<Homotopy invariance of winding number> states that a homotopy $H(s,t)$ through closed paths in $\mathbb C^*$ preserves <winding number>. Uniform continuity divides the $s$ interval so successive paths differ pointwise by less than the minimum <modulus> of the earlier path; the supplied perturbation lemma makes their <winding number>s equal. Chaining the subdivisions proves the result.