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Thread: Examples

  1. April 23rd 2009, 04:59 PM #1
    manjohn12
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    Examples

    Are there any examples that show that the pointwise limit of real-valued bounded functions need not be bounded?

    E.g. f(x) = \lim_{n \to \infty} f_{n}(x) is not bounded if each of the f_{n} 's are bounded?
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  2. April 23rd 2009, 08:15 PM #2
    foxjwill
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    Consider the sequence
    f_n=\begin{cases}<br /> \sum_{k=0}^n x^n, &|\,x\,|<1\\<br /> 0, &|\,x\,|\geq 1.<br /> \end{cases}
    Clearly, each f_n is bounded, yet
    \lim_{n\to\infty}f_n(x)=\begin{cases}<br /> \frac{1}{1-x}, &|\,x\,|<1\\<br /> 0, &|\,x\,|\geq 1.<br /> \end{cases}
    is not.
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