Is there a way to use Mobius transformation to prove existence of Laurent Series

Question

I'd like to prove the existence of a Laurent series for a function with a pole. The kind of proof I have in mind must use an inversion. The key idea is that if $$f\left(z\right)=\sum_{n=-\infty}^\infty a_nz^{n},$$ inside a circle $C_1$ then it must be $$f\left(\frac{1}{z}\right)=\sum_{n=-\infty}^\infty a_n \left(\frac{1}{z}\right)^{n},$$ outside a circle $C_2$. The idea it seems clear but I can't fill in the details... is there some fault in the reasoning?