I need to answer this without using Residue Theorem, Maximum Modulus Principle or more advanced theorems, but rather rely on more basic results such as Cauchy's formulas for integration of analytic functions and the results on upper bounds for moduli of contour integrals.
The way I tried to go about it is:
- First prove that the contour integral of $C_R$ is equal to any contour integral of a circle of a larger radius that contains $C_R$. I know how to do this I believe.
- Showing that $\frac{1}{|P(z)|} < \frac{1}{K|z|^n}$ for all circles of radius big enough, based on a certain lemma for complex polynomials I know how to prove.
- Using the result on upper bounds for moduli of contour integral to then show that the value of the integral of radius big enough is smaller than some negative integer power of its radius.
- Therefore we can keep increasing the radius of the circle we're integrating over to get arbitrarily close to zero, therefore the original integral is equal to zero QED.
Does this sound about right? I'm surprised I haven't found an answer to this since it seems like a fundamental result about polynomials!