How do I prove that f is analytic? Complex functions.

Question

I would like to show that if $F$ is analytic in $A\subset \mathbb{C}$, then so is $f$ where $f =\dfrac{F(z)-F(z_0)}{z-z_0}$ for $z \in A, z \neq z_0$ and $f(z_0)=F'(z_0)$ for $z_0$ some fixed point in A.

I was trying to use Cauchy's integral formula. Maybe? I don't know. I Need Help.