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Thread: Counterexample to uniformly convergence

  1. #1
    Feb 2009
    Valparaíso, Chile

    Counterexample to uniformly convergence

    Hi! My problem is this: Find an example of (f_n), a sequence of functions on \mathcal{C}(X,\mathbb{R}) (continuous with domain X and real-valued) such that: X is NOT compact, (f_n) be equicontinuous and pointwise bounded, and every subsequence uniformly convergent have the same limit (call him f. In fact, may there's no subsequence uniformly convergent, and we aren't saying that every subsequence is uniformly convergent). I need to find a sequence that satisfy this conditions and NOT converges to f uniformly.


    Edit: I think that I don't need your help now =) With f_n(x)=\dfrac{x}{n},\;X=\mathbb{R} holds
    Last edited by Pedro²; Dec 4th 2009 at 01:37 AM.
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