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