This is an exercise from Conway I am stuck at. Here $n($$\gamma$ $,$ $w$) is the winding number and $(6.2)$ is like below.
I need to prove that $h$ is continuous using only the definition of winding number, not Cauchy Theorem. In fact I have to use the result of the exercise 9 to prove the Cauchy Theorem. So, I have to use some $\delta$ $-$ $\epsilon$ argument. But, I can't find a way to do so. Could anyone help me how to solve the exercise 9?
The definition of winding number is like below.
