# Rudin's Proof about Winding Numbers

This is kind of a softball question, an untied loose end that has always bugged me. It is well-known that if $$\Gamma_1\sim \Gamma_2$$ are two homotopic closed paths in a region $$\Omega$$, and if $$\alpha\notin \Omega$$, then $$n(\Gamma_1;\alpha)=n(\Gamma_2;\alpha).$$ I've seen several proofs of this, using approximation by polygonal paths. Rudin's is (surprise!) the slickest, but of course, he leaves some of the details to the reader, and when I do the calculation, I am off by a factor of two at a certain step, which does not affect the proof (one can scale the original hypothesis), but I must be making a mistake, and it has always bugged me. So I'd like to see where my error is.

Let $$H:I\times I\to \Omega$$ be the homotopy. and choose an integer $$n$$ such that

$$|s-t|+|s'-t'|<1/n\Rightarrow$$

$$|H(s)-H(t)|+|H(s')-H(t')|<\epsilon.$$

Define the paths $$\{\gamma_0,\cdots ,\gamma_n\}$$ by

$$\gamma_k(s)=H(i/k,k/n)(ns+1-i)+H((i-1)/n,k/n)(i-ns)$$

if $$i-1\le ns\le i.$$

The claim is then that $$|\gamma_k(s)-H(s,k/n)|<\epsilon.$$

Here is what I am getting, after substituting and applying the triangle inequality:

$$|H(i/n,k/n)-H((i-1)/n,k/n)|(ns-i)+|H(i/n,k/n)-H(s,k/n)|$$

which is easily seen to be $$<2\epsilon.$$ It seems like the only way to avoid the factor of two, would be to arrive at a tractable expression without using the triangle inequality. But I do not see how to do this. Unless at the outset, we should have simply required that

$$|s-t|+|s'-t'|<1/n\Rightarrow$$

$$|H(s)-H(t)|+|H(s')-H(t')|<\epsilon/2.$$