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Thread: Another Royden Question

  1. #1
    Junior Member
    Jun 2010

    Another Royden Question

    Hi all!
    I'm here again with another Royden question. The problem's pretty simple to state:

    Given a sequence of real valued functions: <f_n> where each f_n is continuous and defined for all real numbers, show that
    the set \{x:\ <f_n (x)>\  converges\} is an f_\sigma_\delta

    I'm fairly certain that I've shown that the set is a G_\delta. And I'm pretty sure that I can prove that every G-delta is an F-sigma-delta. This would mean I'm done. But wait! Wouldn't Royden have just asked me to prove it was a G-delta in the first place? By that logic, I think I did something wrong in my proof. Casn anyone lend a hand on this problem. (Or does asnyone think I'm right about the set being a g-delta?) Thanks for any responses. They will be diligently read.

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  2. #2
    Junior Member
    Aug 2010

    here is one way how to write it as a F_{\sigma\delta} set:
    h_{nm}(x):=f_n(x)-f_m(x) is continous, so
    F_{nmr}:=h_{nm}^{-1}([-r,r]) is closed for every r>0
    <f_n(x)> converges iff <f_n(x)> is a cauchy-sequence iff
    \forall r\in \mathbb{Q}^{+} \exists n_{0} \forall n,m\geq n_{0} : x \in F_{nmr}

    this is in F_{\sigma\delta}.

    What is your solution?
    Last edited by Iondor; Aug 1st 2010 at 11:40 AM.
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