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Thread: Examples

  1. #1
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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. $\displaystyle f(x) = \lim_{n \to \infty} f_{n}(x) $ is not bounded if each of the $\displaystyle f_{n} $'s are bounded?
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  2. #2
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    Apr 2009
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    Consider the sequence
    $\displaystyle f_n=\begin{cases}
    \sum_{k=0}^n x^n, &|\,x\,|<1\\
    0, &|\,x\,|\geq 1.
    \end{cases}$
    Clearly, each $\displaystyle f_n$ is bounded, yet
    $\displaystyle \lim_{n\to\infty}f_n(x)=\begin{cases}
    \frac{1}{1-x}, &|\,x\,|<1\\
    0, &|\,x\,|\geq 1.
    \end{cases}$
    is not.
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