Let $\gamma_1, \gamma_2 : [0,1]\to\mathbb{C}$ closed paths and $w\in\mathbb{C}$ a point, so that $$\vert\gamma_2(t)-\gamma_1(t)\vert < \vert\gamma_1(t)-w\vert$$ on $[0,1]$. Prove that $$n(\gamma_1,w)=n(\gamma_2,w)$$ Hint: Use the path $\gamma(t)=\frac{\gamma_2(t)-a}{\gamma_1(t)-a}+a$.
I found some help with homotopy but we actually didn't learned something about this topic. We only get the definition of winding numbers $$n(\Gamma, z_0)=\frac{1}{2\pi i}\int_{\Gamma}\frac{d\zeta}{\zeta-z_0}$$
My first idea was, that I use this definition with $\gamma_2$ and solve the integral to get the line integral for $\gamma_1$. Unfortunately I don't know how to use the inequality given in the task. Any hints? Thank you very much!