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Math Help - Uniform congergence Problem.

  1. #1
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    Uniform congergence Problem.

    I have two question here:
    1) Suppose f_n = x^n ,   \forall x\in [0,1] , n \in $\mathbb{N}$ , how to find A_\varepsilon \subseteq [0,1] such that f_n is converge UNIFORMLY on A_\varepsilon ?

    2) Suppose m(A) \nless \infty ,why there is no such  A_\varepsilon \subseteq A such that when f_n = 1 _{[n,n+1]} , f_n is converges uniformly on A_\varepsilon
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Uniform congergence Problem.

    According to the Dini's Theorem, f_n=x^n\to 0 uniformly on any compact K\subseteq [0,1) .
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  3. #3
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    Re: Uniform congergence Problem.

    Quote Originally Posted by FernandoRevilla View Post
    According to the Dini's Theorem, f_n=x^n\to 0 uniformly on any compact K\subseteq [0,1) .
    Is there any other approach ?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Uniform congergence Problem.

    Quote Originally Posted by younhock View Post
    Is there any other approach ?
    Yes, choose for example K=[0,a) with 0<a<1 , then for every 0<\epsilon<1 and for n positive integer such that n>\log \epsilon/\log a we have n>\log \epsilon/\log x for all x\in K , or equivalently |x^n|<\epsilon for all x\in K .
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  5. #5
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    Re: Uniform congergence Problem.

    Quote Originally Posted by FernandoRevilla View Post
    Yes, choose for example K=[0,a) with 0<a<1 , then for every 0<\epsilon<1 and for n positive integer such that n>\log \epsilon/\log a we have n>\log \epsilon/\log x for all x\in K , or equivalently |x^n|<\epsilon for all x\in K .
    Oh, This is good. Thanks a lot.
    How bout Question 2? I cant show that.
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    Senior Member vincisonfire's Avatar
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    Re: Uniform congergence Problem.

    Quote Originally Posted by younhock View Post
    2) Suppose m(A) \nless \infty ,why there is no such  A_\varepsilon \subseteq A such that when f_n = 1 _{[n,n+1]} , f_n is converges uniformly on A_\varepsilon
    It seems it could be true or false depending on A (if I understand the question properly).
    It would converge uniformly to 0 on [0,1] .
    It would not converge uniformly (but pointwise) to 0 on \bigcup\limits_{n=0}^\infty [n,n+2^{-n}] .
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