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Math Help - Uniform Convergence and Integrals

  1. #1
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    Uniform Convergence and Integrals

    I'm having trouble getting started with this proof. Any help is appreciated...

    If f[0,1]\rightarrow \mathbb{R} is continuous, then, \int _{0}^{1}f=lim_{n\rightarrow \infty}\sum_{k=1}^{n}f(\frac{k}{n})(\frac{1}{n}).

    My thinking is that I need to construct step functions for f but I guess I don't know where to go with that.
    Last edited by zebra2147; March 9th 2011 at 07:24 AM.
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  2. #2
    Senior Member roninpro's Avatar
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    Isn't the term on the right a Riemann sum using rectangles of uniform width 1/n?
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  3. #3
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    So then the height of the rectangle would be defined by f(\frac{x}{n}) since f maps [0,1]\rightarrow \mathbb{R} ?
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