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Math Help - nonobvious limit commuting

  1. #1
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    Unhappy nonobvious limit commuting

    hey all,

    basically I need to justify \lim_{n\to\infty}\lim_{k\to\infty}f_n(z_k)=\lim_{k  \to\infty}\lim_{n\to\infty}f_n(z_k) in this very particular setup

    let f_n(z) be analytic functions on some fixed annulus \{z:r<|z|<R\} where r<1<R. It is given that f_n(z) tend to the constant function 1 uniformly on compact sets of \{z:r<|z|<1\} and of \{z:1<|z|<R\} (so everywhere except of the unit circle). Prove that f_n(z) tend to 1 for z on the unit circle as well.

    WLOG take z=1. we can choose z_k\to 1, |z_k|\ne 1, and so \lim_{n\to\infty}\lim_{k\to\infty}f_n(z_k)=\lim_{k  \to\infty}\lim_{n\to\infty}f_n(z_k) is what we need

    it's clearly crucial that f_n(z) are analytic and converge both inside and outside the unit circle (as the counterexample 1+z^n suggests), but I can't find a way to use it
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  2. #2
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    ok, this is trivial -- easy application of maximum principle

    sorry for bothering
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