Hi. I've been reading Visual Complex Analysis and have been trying to prove Brouwer's Fixed Point theorem on the unit disc as set out in one of the exercises, using winding numbers/Rouche's Theorem. Part 1 I have been able to show, but I am having trouble with proving (or intuitively grassing) part 2 so I was wondering if anyone could help out with this part. Any help welcome! Thank you.
Note: $v[m(C), 0]$ refers to the winding number of $m(c)$ around $0$.
The image of $C_0$ (under the $g(z)-z$ map) is a nonzero point; for small $r$, the image of $C_r$ must be a small loop in the neighborhood of that point, so it cannot wind around the origin. As $r$ increases, the loop grows bigger, but it can never cross the origin (because $g(z)-z$ is never zero), so the winding number can never change.