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Thread: Uniform Convergence

  1. #1
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    Uniform Convergence

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

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

    What we know :
    • Uniform convergence on $\displaystyle (a,b)$ : $\displaystyle \forall \varepsilon>0, \,\exists N_0 \, | \, \forall n \geq N_0, \,\forall x \in (a,b), \, \| f_n(x)-f(x) \| < \varepsilon$
    • Convergence on $\displaystyle x=a$ : $\displaystyle \forall \varepsilon>0, \,\exists N_1 \, | \, \forall n \geq N_1, \, \| f_n(a)-f(a) \| < \varepsilon$
    • Convergence on $\displaystyle x=b$ : $\displaystyle \forall \varepsilon>0, \,\exists N_2 \, | \, \forall n \geq N_2, \, \| f_n(b)-f(b) \| < \varepsilon$
    I suggest you try to build $\displaystyle N$ from $\displaystyle N_0$, $\displaystyle N_1$ and $\displaystyle N_2$.
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  3. #3
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    Okay, that makes things a little clearer. Would it be enough to just let $\displaystyle N=max\{N_0, N_1, N_2\}$?
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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 $\displaystyle f_n$ converge pointwise on $\displaystyle [a,b]$ and uniformly on $\displaystyle [a,b] \setminus \{ c_1,...,c_k\}$ where $\displaystyle c_i\in [a,b]$ then $\displaystyle f_n$ converges uniformly on $\displaystyle [a,b]$.

    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 $\displaystyle \epsilon$ the set of $\displaystyle N$'s for each pointwise point is bounded then the sequence is actually uniformly convergent everywhere.
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