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

    My thinking is that I need to construct step functions for $\displaystyle f$ but I guess I don't know where to go with that.
    Last edited by zebra2147; Mar 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 $\displaystyle 1/n$?
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  3. #3
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    So then the height of the rectangle would be defined by $\displaystyle f(\frac{x}{n})$ since $\displaystyle f$ maps $\displaystyle [0,1]\rightarrow \mathbb{R}$ ?
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