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Another Convergence Question

  1. Jan 25, 2009 #1
    The problem statement, all variables and given/known data
    Let [tex]\{f_n\}[/tex] be a uniformly bounded sequence of functions which are Riemann-integrable on [a,b]. Let
    [tex]F_n(x) = \int_a^x f_n(t) \, dt[/tex]
    Prove that there exists a subsequence of [tex]\{F_n\}[/tex] which converges uniformly on [a,b].

    The attempt at a solution
    I was thinking, since [tex]\{f_n\}[/tex] is uniformly bounded, there is an M such that [tex]F_n(x) \le M(x - a) \le M(b - a)[/tex] for all n, for all x. Now this automatically means that [tex]\{F_n\}[/tex] converges uniformly right? But then if [tex]\{F_n\}[/tex] converges uniformly, why is the problem requesting for a subsequence? I must have done something wrong.
     
  2. jcsd
  3. Jan 25, 2009 #2

    Dick

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    The f_n aren't given to be convergent. Sure the F_n are bounded, but that certainly doesn't mean they are convergent. Can't you think of a theorem that guarantees the existence of a uniformly convergent subsequence in a family of functions? What are it's premises?
     
  4. Jan 25, 2009 #3
    You mean the Arzela-Ascoli Theorem right? The Wikipedia article actually states what I'm trying to prove I think:

    the hypotheses being that the sequence must be uniformly bounded and equicontinuous. I have already shown that the sequence is uniformly bounded right? So all I need to show is that it is equicontinuous. I think I can handle that.
     
  5. Jan 25, 2009 #4

    Dick

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    You've got it.
     
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