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Suppose $f$ is Riemann integrable. Consider a sequence of partitions $P_n$ such that $mesh P_n$ tends to zero as $n \rightarrow \infty$. I want to prove that the sequence of upper sums converge i.e. $\lim_{n \rightarrow \infty}U(f,P_n)$ exists. How to do so ?

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    $\begingroup$ This could be the very definition of f being Riemann integrable. Please make precise which definition you are using. $\endgroup$ – Did Dec 16 '13 at 9:36
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Hint: If $f:[a,b]\to \Bbb R$ a bounded function. Then $f$ is Riemann integrable in $[a,b]$ iff $\forall \epsilon >0$ there is a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<\epsilon$.

Use this in order to prove that $\lim_{n\to \infty} (U(f,P_n)-L(f,P_n))=0$.

Also $f$ is Riemann integrable iff $\underline \int _{a}^{b} f=sup_P(L(f,P))=inf_Q(L(f,Q))=\overline \int_{a}^{b} f$.

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  • $\begingroup$ I understood till $\lim_{n\to \infty} (U(f,P_n)-L(f,P_n))=0$. But how does that prove what I want to prove ? $\endgroup$ – aaaaaa Dec 16 '13 at 11:15
  • $\begingroup$ @Prasenjit Because $f$ is Riemann integrable, $\lim_{n->\infty}U(f,P_n)$ will exist because of the last two lines that i wrote. $\endgroup$ – Haha Dec 16 '13 at 11:55
  • $\begingroup$ Can you elaborate ? $\endgroup$ – aaaaaa Dec 16 '13 at 13:16
  • $\begingroup$ I think this limit exists if the sequence $P_n$ is a increasing sequence (in terms of refinement), otherwise we can't say that the limit exists. My question is regarding that $\endgroup$ – aaaaaa Dec 16 '13 at 13:47

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