Given $\{f_{n}(z)\}$, a sequence of analytic functions in the upper half plane $\mathbb C^{+}$, where each $f_{n}(z)$ has continuous extension to the real line, and $|f_{n}(z)|\leq 1$ for all $z\in \mathbb C^{+}\cup \mathbb R$ (i.e., uniformly bounded). How to prove that there is a subsequence which converges to some analytic function $f$ (whatever the convergence is; uniformly or pointwise)?
(I think there is a theorem called Montel's theorem, but I'm not sure if we can apply it here directly!)