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Math Help - Uniform Convergence

  1. #1
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    Uniform Convergence

    If I know that \{f_n(x)\} converges uniformly on (a,b) and converges pointwise on x=a and x=b, how can I show that \{f_n(x)\} converges uniformly on [a,b]?
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    We want the uniform convergence on [a,b] that is to say : \forall \varepsilon>0, \,\exists N \, | \, \forall n \geq N, \,\forall x \in [a,b], \, \| f_n(x)-f(x) \| < \varepsilon

    What we know :
    • Uniform convergence on (a,b) : \forall \varepsilon>0, \,\exists N_0 \, | \, \forall n \geq N_0, \,\forall x \in (a,b), \, \| f_n(x)-f(x) \| < \varepsilon
    • Convergence on x=a : \forall \varepsilon>0, \,\exists N_1 \, | \, \forall n \geq N_1, \, \| f_n(a)-f(a) \| < \varepsilon
    • Convergence on x=b : \forall \varepsilon>0, \,\exists N_2 \, | \, \forall n \geq N_2, \, \| f_n(b)-f(b) \| < \varepsilon
    I suggest you try to build N from N_0, N_1 and N_2.
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  3. #3
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    Okay, that makes things a little clearer. Would it be enough to just let N=max\{N_0, N_1, N_2\}?
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  4. #4
    Super Member flyingsquirrel's Avatar
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    Yes, it would !
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  5. #5
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    In fact we can strongly generalize this. If f_n converge pointwise on [a,b] and uniformly on [a,b] \setminus \{ c_1,...,c_k\} where c_i\in [a,b] then f_n converges uniformly on [a,b].

    A more interesting question is whether we can have a similar result for infinitely many points removable as well. I am not sure yet.

    EDIT: There is a simple generalization for infinitely many point but it is not an interesting generalization: if for any \epsilon the set of N's for each pointwise point is bounded then the sequence is actually uniformly convergent everywhere.
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