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Showing that there are particular sequences of functions that converge

  1. Sep 10, 2013 #1
    problem_zps819d0945.png

    answer_zpsc11573c9.png


    That seems like a valid argument for showing that [itex]\phi_n[/itex] converges to f, but I'm not sure how to show it's increasing. And as far [itex]\psi_n[/itex], converges, well I imagine that I'd use a similar argument, but I'm still not sure how to show it's decreasing.
     
  2. jcsd
  3. Sep 10, 2013 #2

    Office_Shredder

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    I agree it seems to be missing.

    Since [itex] \phi_n[/itex] is smaller than f, you can define
    [tex] \tilde{\phi_n}(x) = \max_{j=1,...,n} \phi_k(x) [/tex]
    and this will still be within 1/n of f everywhere, and is increasing in n, and each [itex] \tilde{\phi_n}[/itex] is a simple function. Exercise to prove all these properties.
     
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