I have encountered the following problem while reading in complex analysis.
Let $U$ be a domain in $\mathbb{C}$ with $z_0 \in U$. Let $\mathcal{F}$ be the family of analytic functions $f$ in $U$ such that $f(z_0) = -1$ and $f(U) \cap \mathbb{Q}_{\geq 0} = \emptyset$, where $ \mathbb{Q}_{\geq0}$ denotes the set of non-negative rational numbers. Is $\mathcal{F}$ a normal family?
[A normal family of functions is a family such that every sequence of functions from the family has a subsequence which converges uniformly on compact subsets of the domain.]
I know I'm supposed to show my work here, but I don't know how to begin to solve this problem. I know that Montel's Theorem says that if $\mathcal{F}$ is locally uniformly bounded, then it is a normal family, but there doesn't seem to be much to work with to show bounds here. And if it's not a normal family, would I have to produce an explicit sequence which has no uniformly convergent subsequence?
Any guidance would be appreciated. Thanks.