Talk:Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 3

I assume you don't expect me to do anything for the time being. Ivar Sand (talk) 11:01, 28 November 2013 (UTC)

That is correct. We'll get to it eventually. — Lord_Farin (talk) 11:03, 28 November 2013 (UTC)

Background for the proof

Once a friend of mine said that a Cauchy sequence convergences for complete spaces. Moreover, he said that a Cauchy sequence converges for the reals because the space of real numbers is complete. I myself knew nothing about complete spaces since I had only knowledge of elementary real analysis.

However, I felt that his reference to complete spaces should be unnecessary, in other words, that there should be no mandatory logical link between the Cauchy criterion and complete spaces or other advanced concepts. So, I started constructing a proof that the Cauchy criterion and the standard convergence criterion are equivalent using only elementary real analysis. To prove that a convergent sequence is Cauchy is easy, as is well known. The challenging part was to prove that a Cauchy Sequence is convergent. --Ivar Sand (talk) 07:33, 27 November 2018 (EST)

It needs to be pointed out that this proof has a student's perspective. It may not be of interest to the rest of the mathematical community. --Ivar Sand (talk) 10:09, 22 November 2019 (EST)

There is another example of a proof with a student's perspective here (see the discussion page). --Ivar Sand (talk) 04:34, 3 December 2019 (EST)

Suggestions of improvement

I've been looking at this to see why it needs to be so complicated and why you cannot use the Bolzano-Weierstrass Theorem. My question is: is the sticking point the Continuum Property? --prime mover (talk) 02:29, 21 June 2019 (EDT)

I wrote the proof many years ago, in my student days I believe. I wanted to produce a direct proof of the theorem without reference to any advanced concept. I saw the challenge of producing a proof using only my own knowledge of mathematics. I thought that was fun, and I felt satisfied when I succeeded.

But I am aware of the fact that my competence of mathematics is limited even today. I would not be surprised if others with more knowledge of mathematics and who are smarter than I, would be able to change the proof into one that is shorter and better. Therefore, I am not convinced that the proof needs to be so complicated as the one that I produced.

You are mentioning the Bolzano-Weierstrass Theorem. I cannot see exactly how it can be used here, but, as I indicated above, others might understand that better (see proofs 1 and 2).

Regarding whether the sticking point is the Continuum Property (or rather the Monotone Convergence Theorem), yes I suppose it could be. But to me, the essential part of the proof is the construction of the sequences ${U_i}$ and ${L_i}$ and the proof that they converge to the same limit. --Ivar Sand (talk) 03:37, 1 July 2019 (EDT)

The point is that every single step in this proof is developed painstakingly from first principles (and in a very unwieldy way -- what is the point of setting up a sequence of values of $\epsilon_i$ when just stating one value will do the job well enough?) It is seen to be of the same structure as Proof 2, but the latter merely invokes a bunch of previously-demonstrated results which you have proved step by step. One of those results is effectively the B-WT, but as the latter relies upon the Continuum Property and yours does not specifically invoke that, I wondered (sorry, I don't have the patience to check it myself, life's too short) whether you could not actually use B-WT because you specifically did not want to use Continuum Property? --prime mover (talk) 03:51, 1 July 2019 (EDT)

I am sorry, but I believe I misunderstood what you meant by sticking point. I knew the Continuum Property as an axiom of the real number system, so I did not have a problem with that. My problem was connected to the concept of complete spaces since I had not been taught it, so I considered it an advanced concept.

I still cannot see how Proof 3 could benefit from the use of the Bolzano-Weierstrass Theorem. Proof 3 could benefit from the use of the Squeeze Theorem though, but this would not have made it much shorter. --Ivar Sand (talk) 03:45, 3 July 2019 (EDT)

No, sorry, I give up. --prime mover (talk) 04:25, 3 July 2019 (EDT)

I have some more comments, maybe I can clear things up a bit better this time.

The point of setting up a sequence of values of $\epsilon_i$ is, as is explained now at the top of the proof:

The sequence $\sequence {\epsilon_i}$ should be considered a sequence of values or levels that a positive, real variable $\epsilon$ (≥1) passes through on its way towards 0. At each level an upper bound and a lower bound for a subsequence of $\sequence {a_n}$ are defined. All such upper bounds form a sequence of upper bounds and the lower bounds a sequence of lower bounds. These two sequences are crucial for showing that $\sequence {a_n}$ converges.

I don't think that a proof of the Bolzano-Weierstrass Theorem is hidden somewhere in the proof. This is because the two sequences defined in the proof are not subsequences of $\sequence {a_n}$.

Regarding changing of the proof: A central part of the proof is $a_{N_i} - \epsilon_i < a_n < a_{N_i} + \epsilon_i$. I think this part should not be changed so as to keep the nature of the proof intact.

The complexity or the length of the proof is caused by the idea of the proof as expressed by the double inequality mentioned above. For instance, the sequences $\sequence {u_i}$ and $\sequence {l_i}$ are defined from these inequalities and they are indexed differently than $\sequence {a_n}$. This causes the comparison of $\sequence {u_i}$ and $\sequence {l_i}$ with $\sequence {a_n}$ to be complex. --Ivar Sand (talk) 10:40, 15 December 2022 (UTC)

Read Proof 4 instead

I said above "I would not be surprised if others with more knowledge of mathematics and who are smarter than I, would be able to change the proof into one that is shorter and better." Now such a proof is here! Also, it is an elementary proof that does not use advanced concepts. I am talking about Proof 4. Thank you for posting this proof, prime mover. --Ivar Sand (talk) 10:57, 17 November 2022 (UTC)

But I think Proof 4 is not the same approach. Proof 3 looks unique and correct but its exposition is too messy. --Usagiop (talk) 01:30, 18 November 2022 (UTC)

New title for section "$\sequence {a_n}$ is between two other sequences"

Usagiop has suggested (thank you, Usagiop) in an improve template concerning the section title "$\sequence {a_n}$ is between two other sequences": "The title could be more meaningful. Also it is helpful to mention here that you want to construct a sequences decreasing $\sequence {u_n}$ and a increasing $\sequence {l_n}$ such that $l_n \le a_n \le u_n$ and $\lim_{n\to\infty}u_n = \lim_{n\to\infty}l_n$, or something similar."

Note that the title stretches a bit far because in its associated section it is not talked about sequences yet but rather elements of sequences. The thought behind the title is to indicate what is talked about in the following sections as well where sequences are introduced. So the title should have been on a higher level. However, I would like instead to limit the scope of the title to what is talked about in its associated section. My suggestion for a new title: "$a_n$ is between $a_{N_i} - \epsilon_i$ and $a_{N_i} + \epsilon_i$ for a natural number $N_i$".

I intend to explain that the proof is based upon introducing sequences of upper and lower bounds to subsequences of $\sequence {a_n}$ and place that in the beginning of the proof instead of below the section title. --Ivar Sand (talk) 10:56, 23 November 2022 (UTC)

Then the previous title was incorrect and your new suggestion is still imprecise. Please write down the full statement, e.g.:

$\forall i \in \N : \exists N_i \in \N : \forall n \ge N_i : a_{N_i} - \epsilon_i \le a_n \le a_{N_i} + \epsilon_i$

The title could be "Lemma" if nothing better is found. --Usagiop (talk) 13:11, 23 November 2022 (UTC)

The full statement would be too specific in a title in my opinion. The title should be more descriptive than text within the section. Readers who are interested in the whole story should read the associated section. (I may have misunderstood though. Maybe you were talking about full statements within sections in general and not in section titles.)

I don't quite understand what you mean by "Lemma" as a title. I don't think "lemma" is used in section titles, but I may be mistaken. Lemmas are put in separate pages or subpages. --Ivar Sand (talk) 10:24, 25 November 2022 (UTC)

Ignore all the above, what is there is fine. --prime mover (talk) 11:03, 15 December 2022 (UTC)

Another suggestion for a new title: "$a_n$ is between and arbitrarily close to elements of two sequences" or "$a_n$ is between two numbers, $\forall i \in \N : \exists N_i \in \N : \forall n \ge N_i : a_{N_i} - \epsilon_i \le a_n \le a_{N_i} + \epsilon_i$ --Ivar Sand (talk) 16:07, 29 November 2022 (UTC)

I have no strong preference. It is OK. --Usagiop (talk) 18:02, 29 November 2022 (UTC)