Prove region is not simply connected Let $A=\{z=x+iy\in C | 1\lt |z-z_0 | \lt 2 \}$ . ($z_0$ is any point on the plane).
Prove that $A$ is not a simply connected region in $C$.
How can I prove this?  The only idea that comes to my mind is to compute an integral over a closed curve contained in $A$ ,supposing $A$ was simply connected, and then use Cauchy's theorem to get to a contradiction. 
Any help will be appreciatted.

Observe that the index of $\;z_0\;$ wrt the closed, simple curve $\;\gamma: |z-z_0|=\frac32\;$ is not zero, since
$$n(\gamma,z_0):=\frac1{2\pi i}\oint_{\gamma}\frac{dz}{z-z_0}=1$$
and from here that $\;A\;$ isn't simply connected.
