I got the following exercise:
Let $\mathcal{F}$ be the family of all functions $f \in \mathcal{H}(\mathbb{D})$ such that $f = z + a_2 z^2 +a_3 z^3 + \ldots$ with $\vert a_n \vert \leq n$ for all $n \in \mathbb{N}$. Show that $\mathcal{F}$ is a normal family.
My idea so far is the following:
Since $\mathcal{F}$ is a family of polynomials on $\mathbb{D}$ and polynomials are bounded on compact subsets of the complex plane $\mathbb{C}$, Montel's theorem implies that $\mathcal{F}$ is a normal family. I am pretty sure though that this can't be true because I didn't the fact that $\vert a_n \vert \leq n$ anywhere.
Is there a mistake in my thinking or is it simpler than it seems (probably not)
Your argument only shows that single element of $\mathcal F$ gives a normal family.
If $K$ is a compact subset of $\mathbb D$ then $K \subseteq \{z:|z|\leq r\}$ for some $r <1$. The series $1+2r^{2}+3r^{2}+\cdots$ is convergent by ratio test and $ |f(z)|\leq 1+2r^{2}+3r^{2}+\cdots$ for all $z \in K$, for all $f \in \mathcal F$. By Montel's Theorem, $\mathcal F$ is normal.