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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!)

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    Do we care about the boundary? the sequence of functions $x^{n}$ still converge to$0$on $[-1,1]$, right! and this is what we need, on the real line.2012-06-13

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