Let $\Gamma$ denote the positively oriented circle of radius $2$ with center at the origin. Let $f$ be an analytic function on $\{z \in\mathbb{C} : |z|>1 \}$, and let $$\lim_{z\to \infty}f(z)=0.$$
Prove that $$ f(z)=\frac{1}{2\pi i} \int_{\Gamma} \frac{f(\xi)}{z-\xi}d\xi, $$ for all $z\in\mathbb{C}$ with $|z|>2$.
I have no idea how to even start. Any hint will be appreciated.