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I am trying to prove the following:

Suppose $ f:[a,b]\rightarrow\mathbb{R} $ is bounded. Then $ f $ is Riemann integrable if and only if for each sequence of marked partitions $\{P_n\}$ with $\{\mu(P_n)\}\rightarrow0$, the sequence $\{S(P_n,f)\}$ is convergent

,where $\mu(P)$ is the mesh of partition $P$ and $S(P,f)$ is the Riemann sum of $f$ over partition $P$.

My attempt at a solution:

Suppose for each sequence of marked partitions $\{P\}$ with $\{\mu(P_n)\}\rightarrow0,$ $ \{S(P_n,f)\}$ converges.

Let $\epsilon>0$ be given. Then there is an $A\in\mathbb{R}$ and $N\in\mathbb{N}$ such that when $n>N$, there exists $\delta$ such that $\mu(P_n)<\delta\implies |S(P_n,f)-A|<\epsilon$

Then, by the theorem provided by leo below, the existence of $A$ implies that $f$ is Riemann integrable.

Now suppose $f$ is integrable. Then given $\epsilon>0$, there exists $A\in\mathbb{R}$ such that there exists $\delta$ for which $\mu(P)<\delta\implies |S(P,f)-A|<\epsilon, \forall P$.

Then for each sequence each sequence of marked partitions $\{P\}$ with $\{\mu(P_n)\}\rightarrow0,$ $\mu(P_n)<\delta$.

Then, $|S(P_n,f)-A|<\epsilon$ which means that $\{S(P_n,f)\}$ converges to A. Also by the theorem below, $A=\int f dt$

Shafat Arbaz Alam
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    What is $\mu(P_n)$? – leo Feb 01 '12 at 02:13
  • This is a notation I have seen for the mesh of a partition; this is the length of the longest interval in the partition. – ncmathsadist Feb 01 '12 at 02:16
  • Sorry, clarified – Shafat Arbaz Alam Feb 01 '12 at 02:21
  • Use the definition that says: $f$ is integrable over $[a,b]$ if there exist a number $I$ such that for every $\epsilon\gt 0$ exist $\delta\gt 0$ s.t. if $P$ is partition of $[a,b]$ with $\mu(P)\lt\delta$, then $$|S(P_n,f)-I|\lt\epsilon.$$ Now I'm tired. I'll post an answer tomorrow if nobody does. – leo Feb 01 '12 at 03:10
  • I know that definition. I'm curious about how I use the sequential properties to get to there. Thanks for your help! – Shafat Arbaz Alam Feb 01 '12 at 03:19
  • I added some steps, am I on the right track? – Shafat Arbaz Alam Feb 01 '12 at 23:17
  • I think your condition must be: ...for each sequence of marked partitions ${P_n}$ with $\mu(P_n)\to 0$, the sequence $S(P_n,f)$ converges to $A$. I mean, you need that all such sequences converges to the same limit. – leo Feb 02 '12 at 05:23
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    Converges is enough: if two sequences converge to some different limits, interlace them to get a divergent sequence, which contradicts the hypothesis. – Did Feb 25 '12 at 12:02
  • Check the reference in my answer? If you give those definitions, I can write the proof for the corollary. (not to the main theorem though.) Meanwhile, want to look at my related question? -- http://math.stackexchange.com/questions/1803080/if-the-left-riemann-sum-of-a-function-converges-is-the-function-integrable – Argyll May 28 '16 at 06:58
  • Just to be clear, I don't mean my question will help you. I was hoping for answers to my question. Though it seems like my question is getting good attention as I type. – Argyll May 28 '16 at 07:06

1 Answers1

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Your statement should be an easy corollary of the Du bois-Reymond and Darboux integration theorem. The proof of the theorem is rather cumbersome. So here is a reference: the proof can be found in Analysis by Its History by Hairer and Wanner.

And by the way, you need to define what convergence as $\{\mu(P_n)\}\to 0$ and what marked partition mean.

Argyll
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