I'm self-studying complex analysis, and in my book there are starred exercises on complex integration I'm interested in understanding.
Lemma 1 of the text states
If the piecewise differentiable closed curve $\gamma$ does not pass through the point $a$, then the value of the integral $$ \int_\gamma\frac{dz}{z-a} $$ is a multiple of $2\pi i$
in preparation for defining the winding number.
One exercise says, give an alternate proof of Lemma 1 by dividing $\gamma$ into a finite number of subarcs such that there exists a single-valued branch of $\text{arg}(z-a)$ on each subarc. Pay particular attention to the compactness argument needed to prove the existence of such a subdivision.
I thought about it a bit, and don't really know how to approach it. Is there a proof or possibly a sketch I could attempt to work through in the meantime? Thank you.