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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. $

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