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Math Help - Convergence in compact Hausdorff space

  1. #1
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    Convergence in compact Hausdorff space

    Hi,
    I've problem with following convergence (in red shape). The second one term is tends to zero it is obvious but why the first one is tends to zero?
    Thanks for any help.


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  2. #2
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    You are told that x_n\to x in the sup norm. This means that, given \varepsilon>0, there exists N such that n\geqslant N\ \Rightarrow\ \|x_n-x\|_{\text{sup}} = \sup\{|x_n(t) - x(t)|:t\in X\} <\varepsilon. Take t=p_n to see that n\geqslant N\ \Rightarrow\ |x_n(p_n) - x(p_n)|<\varepsilon. Thus |x_n(p_n) - x(p_n)|\to0 as n\to\infty.
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