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Math Help - Uniform Convergence

  1. #1
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    Uniform Convergence

    Am I right in saying that if I have f_n=\frac{x^n}{x^n+1} and

    f(x) = \left\{\begin{array}{c l}  0 & x \in [0,1)\\  0.5 & x =1         \\  1   & x\in(1,\infty)\end{array}\right.

    definied on (0,\infty) is not uniform convergent to f(x) (the problem being near the 1)?

    Thanks for any help
    Last edited by Plato; October 19th 2011 at 03:16 AM. Reason: LaTeX fix
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  2. #2
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    Re: Uniform Convergence

    According to the uniform limit theorem, the uniform limit of a sequence of continuous functions is continuous, so you are right.
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  3. #3
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    Re: Uniform Convergence

    Cool thanks (I have proved it from the definition, which is a lot longer, feel a bit foolish now for asking!) it is pointwise convergent to f(x) though right?
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  4. #4
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    Re: Uniform Convergence

    Yes.

    Hint: To avoid <br/> in LaTeX formulas, remove all line breaks between [tex] and [/tex] (not LaTeX line breaks \\; just put the whole formula on one editor line).
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  5. #5
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    Re: Uniform Convergence

    Thanks very much for all the help (sorry about the poor LaTex and thanks for editing it)
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  6. #6
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    Re: Uniform Convergence

    Ha, it was Plato who edited it!
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