Let w be a complex number. Let y, x: [0,1] to C be closed curves such that for all t in [0,1], |y(t)-x(t)| < |y(t) -w|.
By computing the winding number n(s,0) of the closed curve s(t) = (x(t) -w) / (y(t) -w) about the origin, show that n(y,w)=n(x,w).
I do understand the winding number and the result seems obvious, i am just unsure how to do it rigourosly.
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