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Showing that there are particular sequences of functions that converge

  • Thread starter jdinatale
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  • #1
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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.
 

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  • #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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