Let $\mathcal{F}$ denote all analytic functions $f:\mathbb{D} \to \mathbb{C}$ satisfying the inequality $|f(z)| \leq \frac{1}{(1-|z|)^{2015}}$ for all $z\in \mathbb{D}$. Show that $\mathcal{F}$ is a normal family.
My attempt: Let $K\subset \mathbb{D}$ be an arbitrary compact set. There exists $0 <r < 1$ such that for all $z \in K$, $|z| < r$. For every $f \in \mathcal{F}$, and every $z \in K$ we have, $$|f(z)| \leq \frac{1}{(1-|z|)^{2015}} < \frac{1}{(1-r)^{2015}} < \infty.$$ Therefore by Montel's theorem, $\mathcal{F}$ is a normal family.
Is this correct? This is a question from an old qualifying exam, so I'm worried this solution is too short and I might be missing something.