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Pointwise convergence

  1. Nov 23, 2011 #1
    1. The problem statement, all variables and given/known data

    Suppose a sequence [itex](f_n)_{n\in\mathbb{N}}[/itex] converges to a limit [itex]f[/itex] in the metric space [itex](C[a,b],d_{\infty})[/itex] (continuous real valued functions on the interval [a,b] with the uniform metric.)

    Show that [itex]f_n[/itex] also converges pointwise to [itex]f[/itex]; that is for each [itex]t\in [a,b][/itex] we have [itex]f_n(t)\to f(t)[/itex] in [itex]\mathbb{R}[/itex].

    2. Relevant equations

    Uniform metric: [tex]d_{\infty} (f,g) = \text{max}_{t\in [a,b]} |f(t)-g(t)|[/tex]
    3. The attempt at a solution

    [itex]f_n \to f[/itex] in [itex](C[a,b],d_{\infty}) \iff d_{\infty}(f_n,f)\to 0[/itex]

    [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff \text{max}_{t\in [a,b]} |f_n(t)-f(t)| \to 0[/itex]

    [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff|f_n(t)-f(t)| \to 0[/itex] for all [itex]t\in [a,b][/itex]

    [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff f_n(t) \to f(t)[/itex] for all [itex]t\in [a,b][/itex]

    Does this prove it?
     
    Last edited: Nov 23, 2011
  2. jcsd
  3. Nov 23, 2011 #2

    Dick

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    Looks ok to me.
     
  4. Nov 23, 2011 #3

    LCKurtz

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    You need to be careful what you are proving. You have all these [itex]\iff[/itex] implications which would lead one to believe that pointwise convergence and uniform convergence are the same. But they aren't. So look at your argument carefully and make sure the implications go in the direction to prove what you want to prove.
     
  5. Nov 23, 2011 #4

    Dick

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    Good point!
     
  6. Nov 24, 2011 #5
    Obviously to prove what I want I only need all steps to be [itex]\implies[/itex] but which step above is not "if and only if"? (is it the last step?)
     
  7. Nov 24, 2011 #6

    Dick

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    |f_n(t)-f(t)|->0 for all t does not imply max |f_n(t)-f(t)|->0. Can you think of a counterexample?
     
  8. Nov 24, 2011 #7

    LCKurtz

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    And I would add that if I were handing in a proof, I would use a tighter argument. While your implications in one direction are OK, you wouldn't have made that mistake if your argument went something like:

    [tex]0 \le |f_n(t)-f(t)| \le ... \rightarrow 0[/tex]

    where you fill in the dots with reasons for each step.
     
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