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Thread: Cauchy Criterion

  1. #1
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    Cauchy Criterion

    Theorem. If $\displaystyle f $ is Riemann integrable then its Riemann sums satisfy the Cauchy Criterion for the existence of the integral.

    Cauchy Criterion: For every $\displaystyle \epsilon >0 $, there exists $\displaystyle \delta >0 $ such that if $\displaystyle P $ is a partition of $\displaystyle [a,b] $, with $\displaystyle ||P|| < \delta $ and $\displaystyle \mathcal{R}_{1}(f,P) $ and $\displaystyle \mathcal{R}_{2}(f,P) $ are Riemann sums of $\displaystyle f $ on $\displaystyle P $, then $\displaystyle |\mathcal{R}_{1}(f,P)-\mathcal{R}_{2}(f,P)| < \epsilon $.

    Proof. Suppose $\displaystyle f $ is Riemann integrable on $\displaystyle [a,b] $. Then for all $\displaystyle \epsilon>0 $, there exists $\displaystyle \delta >0 $ such that $\displaystyle |\mathcal{R}(f,P)-I| < \epsilon $ whenever $\displaystyle \mathcal{R}(f,P) $ is a Riemann sum for $\displaystyle f $ corresponding to a partition of $\displaystyle [a,b] $. Thus $\displaystyle |\mathcal{R}_{1}(f,P)-I| < \epsilon/2 $ and $\displaystyle |\mathcal{R}_{2}(f,P)-I| < \epsilon/2$. Hence $\displaystyle |\mathcal{R}_{1}(f,P)- \mathcal{R}_{2}(f,P)| < \epsilon $. $\displaystyle \diamond $

    Correct?

    Also could we replace $\displaystyle \mathcal{R}_{1}(f,P) $ and $\displaystyle \mathcal{R}_{2}(f,P) $ with $\displaystyle \mathcal{U}(f,P) $ and $\displaystyle \mathcal{L}(f,P) $. And so the inequality would then be: $\displaystyle |\mathcal{U}(f,P)-\mathcal{L}(f,P)| < \epsilon $? Because these are not necessarily Riemann sums.
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  2. #2
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    Basically $\displaystyle \epsilon/2 $ was arbitrary.
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