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Can you please help me proving , uniform convergence implies pointwise convergence

  1. Oct 7, 2009 #1
    I don't know, I started from the definition of uniform convergence and it seems pretty obvious to me , can anybody start me at least towards right direction?
     
  2. jcsd
  3. Oct 7, 2009 #2

    HallsofIvy

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    Re: can you please help me proving , uniform convergence implies pointwise convergenc

    Yes, that pretty much all it takes.

    Definition of "uniformly convergent": [itex]\{f_n(x)\}[/itex] converges to f(x) as n goes to infinity uniformly if and only if, given [itex]\epsilon> 0[/itex], there exist N such that if n > N, then [itex]|f_n(x)- f(x)|< \epsilon[/itex].

    Definition of "convergent": [itex]{f_n(x)}[/itex] converges to f(a) for given a as n goes to infinity if and only if given [itex]\epsilon> 0[/itex], there exsit N such that if n> N, then [itex]|f_n(a)- f(a)|< \epsilon[/itex].

    Basically, "uniformly convergent" requires that, given [itex]\epsilon[/itex], you be able to use the same [itex]\delta[/itex] for every value of x. Just "convergent" means the value of [itex]\delta[/itex] may depend upon [itex]\epsilon[/itex] and the value of x at which the function is evaluated.
     
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