Prove the family of function is compact & normal

Question

I have a problem with proving something.
I know that "a family $\mathcal{F}$ of function is called compact normal family if the uniform limits of all sequences converging in $\mathcal{F}$ are themselves member of $\mathcal{F}$."

Here a family of functions $\mathcal{F}=\{ f(z)=\sum_{n=0}^{\infty} a_n z^n \ |\ \ |a_n |\leq n \}$ defined in the disk of $|z|<1$ and $z \in \mathbb{C}$.

I want prove this family $\mathcal{F}$ is compact normal. I know that this family is normal because $|f(z)| \leq \sum_{n=0}^{\infty} |nz^n| $ and by Weistrass's M test, it's absolutely and uniformly converge and bounded. So, $\mathcal{F}$ is normal by Montel's theorem.
But i can't about compact..

I can't understand 3 things.

1. What does this family's "sequences" look like?
2. At this time, how should we prove this sequence's uniform convergence?
3. Where should i start with the proof? I wish someone could give me a hint.