Show that $Z^n$ for a nonnengative integer $n$ is normal family

Question

Show that the function $z^n$ for a nonnegative integer $n$ form a normal family in $|z|<1$ ,also in $|z|>1$ but not in the region that contains a point on the unit circle.

I am working on this problem from Complex Analysis by Lars Ahlfors and I need help. This is what I am thinking, And it's a bit vague. Since the family $Z^n, n\geq 1$ converges for all $z$ within the unit circle, Any subsequence of $z^n$ converges. on the other hand, considering $|z|>1$ I don't know how to approach it. Any assistance will be appreciated. Thank you

Answer 1

A family $\mathcal{F}$ of holomorphic functions on a region $\Omega$ is normal if and only if $\mathcal{F}$ is uniformly bounded on compact subsets.

In your case $z^n$ is uniformly bounded in the unit disc, and in $|z|>1$ every sequence contains a subsequence which tends uniformly to $\infty$. We also consider this latter case normal. See Ahlfors Definition 3:

A family of analytic functions is normal if every sequence contains a subsequence converging on compact subsets or a subsequence that tends to $\infty$ uniformly

Can you see what happens when $|z|=1$?