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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: $\displaystyle <f_n>$ where each $\displaystyle f_n$ is continuous and defined for all real numbers, show that
    the set $\displaystyle \{x:\ <f_n (x)>\ converges\}$ is an $\displaystyle f_\sigma_\delta$

    I'm fairly certain that I've shown that the set is a $\displaystyle G_\delta$. 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 $\displaystyle F_{\sigma\delta}$ set:
    $\displaystyle h_{nm}(x):=f_n(x)-f_m(x)$ is continous, so
    $\displaystyle F_{nmr}:=h_{nm}^{-1}([-r,r])$ is closed for every $\displaystyle r>0$
    $\displaystyle <f_n(x)> $converges iff $\displaystyle <f_n(x)>$ is a cauchy-sequence iff
    $\displaystyle \forall r\in \mathbb{Q}^{+} \exists n_{0} \forall n,m\geq n_{0} : x \in F_{nmr} $

    this is in $\displaystyle F_{\sigma\delta}$.

    What is your solution?
    Last edited by Iondor; Aug 1st 2010 at 10:40 AM.
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