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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 $\displaystyle (f_n)$, a sequence of functions on $\displaystyle \mathcal{C}(X,\mathbb{R})$ (continuous with domain $\displaystyle X$ and real-valued) such that: $\displaystyle X$ is NOT compact, $\displaystyle (f_n)$ be equicontinuous and pointwise bounded, and every subsequence uniformly convergent have the same limit (call him $\displaystyle f$. 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 $\displaystyle f_n(x)=\dfrac{x}{n},\;X=\mathbb{R}$ holds
    Last edited by Pedro²; Dec 4th 2009 at 12:37 AM.
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