Let $\displaystyle f(z)$ be analytic on the punctured disk $\displaystyle \{ 0 < |z| < 1 \}$, and define $\displaystyle f_n(z)=f(\frac{z}{n})$, $\displaystyle n \geq 1$. Show that $\displaystyle \{ f_n(z) \}$ is a normal family on the punctured disk if and only if the singularity of $\displaystyle f(z)$ at $\displaystyle 0$ is removable or a pole.

I am having a hard time proving this. I am also not sure if either direction is trivial. I would appreciate some hints on this problem. Thanks.