A sequential, if-and-only-if criterion of Riemann integrability that follows from the Riemann’s definition of the integral, states that a function on a closed and bounded interval is Riemann integrable if and only if any respective sequence of Riemann sums is Cauchy for any tagged partition of the interval with norm tending to zero [1].
Although this is not a new result, it is not mentioned in standard textbooks [2-5].
However, if the previous criterion is used as a starting point to define the integral, it provides a number of educational advantages over the traditional definitions of Riemann and Darboux.
The uniqueness, the linearity, and the order properties of the integral all follow naturally and easily from the respective properties of sequence limit [1].
Moreover, the boundedness of any integrable function is not necessary to be assumed beforehand, as it can be easily proved using that a convergent sequence is bounded [1].
Further, this approach has the advantage of being “infinitesimal-friendly” (so to speak), as the integral can be realized as an infinite sum of infinitesimal quantities [6], thus giving the instructor the opportunity to further introduce the concept of infinitesimal.
Appealing mainly to mathematicians, I wonder if a work presenting such an approach would be publishable in educationally-oriented journals of mathematics.
[1] Presentation The Riemann Integral via Cauchy Sequences of Riemann Sums
[2] Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis. John Wiley & Sons, Inc., Fourth Edition, 2011.
[3] Stephen Abbott, Understanding Analysis. Springer, Second Edition, 2015.
[4] Michael Spivak, Calculus. Publish or Perish, Inc., Third Edition, 1994.
[5] Walter Rudin, Principles of Mathematical Analysis. McGraw-Hill, Inc., Third Edition, 1976.
[6] Preprint A Sequential Proof of the Additivity Property of the Riemann Integral
Dear Spiros,
Many books in elementary mathematical analysis contain sequential approach of Riemann integrability. It is very surprising for me that you say the sequential approach would be a novelty as appearance in elementary analysis books! Clearly, I refer to the correct sequential approach.
Your approach with Cauchy sequences is not correct, the second part of proof of your main theorem in [1] contains errors. It is not sufficient that all sequences S(f;Pn) where //Pn//->0 be Cauchy sequences. The uniqueness of limit of all these sequences is not ensured.
The unique correct sequential approach is:
THEOREM f:[a,b]->R is Riemann integrable if and only if it exists a real number L so that every sequence of the form S(f;Pn) where //Pn//->0 converges to L.
S(f;Pn)=sumi=1,k(n)(xi,n-xi-1,n)f(ti,n) where Pn=(a=x0,n
I do not think that you have read the proof.
You say that the uniqueness of limit of all Cauchy sequences of Riemann sums is not ensured, but it is ensured, because if two such sequences converge to different limits, then combining the two respective sequences of tagged partitions in a way that the odd-numbered terms of the new sequence of tagged partitions is the first old sequence of tagged partitions and the even-numbered terms of the new sequence of tagged partitions is the second old sequence of tagged partitions, you obtain a new sequence of tagged partitions with norm tending to zero and at the same time, the respective sequence of Riemann sums diverges, because its subsequences of odd-numbered and even-numbered terms, respectively, converge to different limits; hence the sequence is not Cauchy.
If this is not clear, I will provide more details.
Actually, I prove this in section 3, "Common limit", by direct proof.
The theorem you present is mentioned in some textbooks, but it is a version stronger than mine.
Spiros Konstantogiannis
Your confusion comes from the approach of tagged partitions.
If we have two partitions sequences (Pn)n and (Qn)n of [a,b], we can obtain another partitions sequence (Rn)n, joining Pn and Qn.
Pn=(a=x0,n
I do not join the partitions, I take them separately, in turn.
The point 2 in your reasoning does not hold in this case.
I explain, once more.
Let (Pn), (Qn) be two sequences of tagged partitions of [a,b] such that the respective sequences of norms both converge to zero.
We consider the sequence (Rn) of tagged partitions of [a,b] be defined as follows:
R1=P1, R2=Q1, R3=P2, R4=Q2 , R5=P3, R6=Q4, etc.
Is not true that (Rn) is also a sequence of tagged partitions of [a,b] with the respective sequence of its norms tending also to zero, as a consequence of the fact that the sequences of norms of both (Pn), (Qn) converge to zero?
Is not true that the sequence of Riemann sums (S(f;Rn)) is such that
S(f;R2n-1)=S(f;Pn) and S(f;R2n)=S(f;Qn)?
Spiros Konstantogiannis
Please write explicitly the formula of partition Rn and formula of sequence S(f;Rn) for all n, taking into consideration the notations provided by me for tagged partitions Pn and Qn! Above, I used standard notations, not incorrectly notations for partitions sequences, like those used in your preprints.
Please let me know why your papers are not published in some peer reviewed journals? At least in Mathematics Magazine or American Mathmatical Monthly!
I use the notation used in Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis. John Wiley & Sons, Inc., Fourth Edition, 2011.
This is a standard and well-known real analysis textbook that is taught in many universities in USA and worldwide.
I use this book as main reference, since it focuses on the Riemann's definition of the integral, and Riemann's definition is my starting point.
I suppose you know the book, and since you mention the American Mathematical Monthly, you may be aware of the celebrated paper "Return to the Riemann Integral", publushed by Robert G. Bartle in 1997 on this journal.
So, the notation I use is standard notation, your claim is totally baseless.
I'm not a mathematician, I'm a physicist.
I have published four peer-reviewed articles on mathematical physics; you may find them in my profile.
Spiros Konstantogiannis
Then send to professor Bartle your manuscripts regarding Riemann integrability, to evaluate the correctness of your statements and proofs! I am convinced that he will not refuse you and will give you the right answer!
Also, send your manuscripts to a peer reviewed mathematical journal, for publication. You will receive the suitable answer! I think you know what this will be!
If you are not a mathematician, why do you think that things that great mathematicians of the 20th century did not discover, you can discover? It is impertinent to combine your preprints with books written by reputed mathematicians such as Bartle, Michael Spivak or Walter Rudin!
Yes, my way to approach our discussion was no more a scientific way, after finding that you don't understand the correct scientific path! I don't understand how certain people end up giving opinions about things they don't master perfectly!
No more words, no more messages! Good luck!
You have been heard. Now let us leave any other person who wishes to express his/her opinion to be heard too.
For convenience to those who wish to answer, I attach the lemma, which all the fuss is about, and a proof by contradiction.
Dear Spriros,
You are right. I appreciate your efforts.
Dinu has not been correct both mathematically
and morally.
In your paper, instead of "Cauchy" you have to write
"convergent".
"Cauchy " is misleading now. Moreover, in general, it
does not imply "convergent".
Note that [a, a]={a}, in contrast your statement " is ill-defined " .
Both you and Dinu are using very bad notations concerning divisions.
Some better notations can be found in my enclosed papers.
With best regards,
Arpad Szaz
Dear Arpad Szaz ,
thank you very much, for your post and your suggestions.
Dear All,
In an attempt to resolve the issue, I submitted a note to the American Mathematical Monthly containing the lemma which DT believes is wrong, both its statement and its proof.
The lemma and the proof are contained in my presentation and they are also attached in a previous post of mine.
I disclose below the remarks I received as they have been sent to me, but first I should say that even though reviewers’ comments are considered confidential, I decided, after much thought, to disclose them for educational reasons, as this will (hopefully) help DT to understand that he has been badly mistaken, primarily mathematically
“This manuscript asks and answers a hypothetical question: Suppose we have a real function on a closed and bounded interval, and every sequence of tagged partitions with mesh approaching zero yields a convergent Riemann sum, is it possible that there is more than one such Riemann sum among these limits? The answer is no, and the proof is exactly what one might expect: take two sequences of such tagged partitions yielding two different Riemann sums, and interleave them. This idea is of the level of an exercise in an undergraduate analysis course, but it is not suitable for publication in the Monthly”.
Dear Spriros,
The last sentence of the referee can be accepted.
However, the former ones are evidently meaningless.
With best regards,
Arpad Szaz
Dear Prof. Arpad Szaz ,
I did not expect that they would publish it; since it is elementary.
However, I submitted it to receive one more credible feedback, like yours, in order to demonstrate that DT has been badly mistaken, primarily mathematically.
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