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Remembering Prof. Jack Huizenga's retracted critiques of the New Calculus

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John Gabriel

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Sep 6, 2017, 2:25:11 AM9/6/17
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Jackie caused a tremendous amount of attention to be focused on my New Calculus with his scorching critique. After I refuted him, he realised he had been impulsive and retracted most of his lies and libel.

Alas, the damage had been done.

9/21/2016 A Harvard alumnus comments! | The New Calculus

Just a few weeks ago, some ignoramus posted a question on that ridiculous site Quora.

The question was:

“What do people think of John Gabriel’s New Calculus, which claims it is the only rigorous formulation of calculus”.

Quora is a question and answer site, that is run by a group of mainstream academics and their sock puppets. They control both the questions and answers. Those who post questions or comments with opposing views, have both their questions and comments mercilessly edited, and deleted if need be. In a sense, it’s Wikipedia-esque, in that the questions and answers are the views of mainstream academics. Nothing that is different or new is tolerated.

One of the fools who sits atop the trash heap, is one Prof. Jack Huizenga, a Harvard alumnus and mathematics teacher at Illinois University. The following link contains a snapshot of Huizenga’s comment: The ridiculous comment at the ridiculous site Quora.

In this very first post, I will address the comments made by this ignoramus (Huizenga) and horrify those, who will realise that this dimwit actually teaches mathematics!

There has been a completely rigorous notion of limit for well over a hundred years. – Huizenga

The great mathematics historian Carl Boyer had this to say:
"Cauchy had stated in his Cours d'analyse that irrational numbers are to be regarded as the limits of sequences of rational numbers. Since a limit is defined as a number to which the terms of the sequence approach in such a way that ultimately the difference between this number and the terms of the sequence can be made less than any given number, the existence of the irrational number depends, in the definition of limit, upon the known existence, and hence the prior definition, of the very quantity whose definition is being attempted.

That is, one cannot define the number 'square root of 2', as the limit of the sequence 1, 1.4, 1.41,1.414, ... because to prove that this sequence has a limit one must assume, in view of the definitions of limits and convergence, the existence of this number as previously demonstrated or defined. Cauchy appears not to have noticed the circularity of the reasoning in this connection, but tacitly assumed that every sequence converging within itself has a limit." - The History of Calculus and its Conceptual Development' (Page. 281) Carl B. Boyer

I doubt that Huizenga ever read this book, and if he did, there is no doubt he did not understand it. Not only has the notion of limit never been rigorous, but it was never challenged by any intelligent mathematician.

Rigorous treatments of infinitesimals are a bit more tricky, but have also been made. – Huizenga

One can only assume the moron is referring to the abortion of non-standard analysis, by a failed Jewish mathematician called Abraham Robinson. Even today, there are many academics (including PhDs) who do not accept non-standard analysis. In my opinion, it is pure rot because infinitesimals
don’t exist.

Reformulations of calculus as attempted in New Calculus, are essentially no interest to mathematicians, as the field of calculus is already extremely well understood and rigorous. – Huizenga

The buffoon wrongly assumes that the New Calculus is only a reformulation, which is evidently false. He goes on to talk about what interests mathematicians, but how could he know? Huizenga is not a mathematician, he is a teacher with no great works behind his name, unless of course one calls his juvenile papers on algebraic geometry a work of any kind.

Needless to say, if calculus were already extremely well understood, we would not have other mathematics PhDs making statements as follows:

Clearly, our calculus course does not prepare scientists in other fields to recognize, understand, and utilize the calculus that many of their fields are based upon. Thus, when it comes to calculus, we don’t get it the first time around, our colleagues don’t get it, and our students are still not getting
it. It’s no wonder that one of the most common occurrences in higher education is that of a non mathematics faculty member discovering that something they were doing is calculus. And at the very least, we feel justified in asserting that there still is a crisis in calculus instruction. – Knisley (Crisis in calculus education)

Knisley goes on to say:

However, in the calculus curriculum, many of the associations are circular. All too often a given concept is associated with a concept that is defined in terms of the original concept. Such connections increase the complexity of a concept without shedding any insight on the concept itself. Not surprisingly, concepts motivated with circular associations are the ones most often memorized with little or no comprehension. – Knisley (Facing the crisis in calculus education)

As for calculus being rigorous, we will hold off on that one until you have read and studied my New Calculus.

There is no point in trying to remove limits or infinitesimals from a discussion of calculus. – Huizenga

It’s easy to see that our moron academic knows nothing about Newton’s calculus and how he (Newton) arrived at the knowledge of it. The author’s main objection to the standard treatment of calculus via real analysis seems to be that he does not understand it. – Huizenga

Well, that’s certainly news to me. I have never made such a claim on the internet or anywhere else. I reject real analysis, not because I don’t understand it (few can ever understand it as well as I), but because it deals with a topic about a non-existent concept – the real number. Real numbers do not exist, because irrational numbers do not exist. In fact, until I arrived on the scene, no one before me and after Euclid, understood what is a number.

In the following comment I debunk the concept of Dedekind cuts and Cauchy sequences:

https://drive.google.com/open?id=0B-mOEooW03iLSTROakNyVXlQUEU

Our buffoon continues:

The errors in New Calculus are too numerous for it, to be worth going through the whole text. – Huizenga

It’s rather funny that he should say this, because he goes on to show that there is not a single error. Unsurprisingly, the baboon dismissed the rest of the text because he could not understand it up till that point. Huizenga mentions that one has to consider an (m,n) pair which is outright false.

In fact, if he had only continued to study and reread the text carefully, he would have soon realised that the values of m and n play no role in that of the gradient. Tsk, tsk.

His definition therefore presupposes that we know the slope of the tangent line (and that we have a notion of a tangent line to a graph of an arbitrary function) and merely computes the slope of a parallel secant line. This is incredibly circular. – Huizenga

Of course we know the slope of the tangent line, provided we know the slope of a parallel secant line. This is grade 8 mathematics! What he states in parenthesis is even more amusing. It demonstrates clearly that he, like many of his colleagues never understood calculus:

If a given function is not continuous and smooth, then any of the methods of calculus are null and void.

One, and only one tangent line exists at every point, provided the function is continuous and smooth, and a given point is not a point of inflection, that is, only half-tangent lines are possible at points of inflection. Our moron then continues to quote the example of the cubic, where there is no
tangent line at x=0 (because an inflection point exists at x=0). Unlike Newton's flawed formulation, the New Calculus handles this correctly.

He further goes to great length to try and convince the reader that he can actually divide the numerator in his difference quotient by m+n. – Huizenga

I don’t go to any great lengths. The proof that every term in the numerator has a factor of m+n can be shown by a high school student. It requires no special knowledge. The first page of the New Calculus website http://thenewcalculus.weebly.com has many examples on this, and includes links
to dynamic Geogebra applets which prove conclusively that it is based on sound analytic geometry.

It is legal to do this in the New Calculus, but illegal in Cauchy's flawed formulation.

That m+n is a factor of every term in the numerator finite difference f(x+n)-f(x-m) can be seen by anyone with a modicum of intelligence, but our chump Huizenga lacks even this. The chump way to see this fact, is to investigate actual functions using the finite difference, and it is soon realised, that this fact is true for all functions. It gets slightly more complicated in the case of terms containing only m and/or n, but this is easy to prove by mathematical induction and constructive proofs, as I have shown in my article

https://drive.google.com/open?id=0B-mOEooW03iLWldTU1ZkTDVQR0E

The easiest proof is given with the equation of a straight line, say f(x)=kx+p. We have from the New Calculus derivative definition: f ' (x) = {k(x+n)+p - [k(x-m)+p] } / (m+n) =k(m+n) / (m+n) = k. Observe that m and n play no role in the value of k which is the gradient. What amuses me, is that so many grade 8 students understand this, and most PhD chumps just don't get it!

Of course the New Calculus is likely to be revised over time to address concerns brought up by people. As this happens, I am confident it will look more and more like standard calculus (or the theory of the symmetric derivative which is closely related). – Huizenga

Another presumptuous claim by our buffoon Huizenga, because the New Calculus has never been revised and there are no plans to revise it whatsoever. There is no need to revise theory that is based on well-defined concepts. It will stand the test of time, just as Euclid’s Elements did.

The statement in parenthesis is quite amusing because it once again demonstrates the lack of understanding displayed by Huizenga. The New Calculus is not just about a new derivative definition, but also about a new integral definition, and much more (*). He might have gotten to that information had he continued studying the text. But ‘open-minded’ academic that he is not, he
ceased to continue, when he could no longer understand what he was reading.

The new calculus derivative definition has nothing in common with the symmetric derivative which Huizenga clearly does not understand. The symmetric derivative requires no special relationship exist between m and n. In fact, the symmetric derivative is used mostly in numeric differentiation.

(*) There are many new methods and theorems in the New Calculus that are not possible using Newton's flawed formulation.

And that covers his comment at Quora. In fact, he proves by all his statements, that there are no errors in the New Calculus, only serious issues in his ability to comprehend.

This same moron thinks that the derivative of sin(x) is not always cos(x). He also fancies that the sine function can take degrees as input, which is outright false. The trigonometric ratios operate only on radian input.

Comments are unwelcome and will be ignored.

Posted on this newsgroup in the interests of public education and to eradicate ignorance and stupidity from mainstream mythmatics.

gils...@gmail.com (MIT)
huiz...@psu.edu (HARVARD)
and...@mit.edu (MIT)
david....@math.okstate.edu (David Ullrich)
djo...@clarku.edu
mar...@gmail.com

Markus Klyver

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Sep 6, 2017, 12:47:54 PM9/6/17
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But you don't understand the Cauchy construction The square root of 2 is not defined as some mystical limit. A such approach is circular. Instead we define a real number as an equivalence class of Cauchy sequences.

John Gabriel

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Sep 8, 2017, 6:44:24 PM9/8/17
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Moron. You need to understand. And it is because of misguided fools like you that I wrote the article

https://drive.google.com/open?id=0B-mOEooW03iLSTROakNyVXlQUEU

Markus Klyver

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Sep 9, 2017, 6:39:10 AM9/9/17
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But you are wrong. We don't define pi in terms of pi. In the case of the Cauchy model, every real number is an equivalence class of Cauchy sequences. An an equivalence class is a set. In the case of the Dedekind construction, a real number is a set of rational numbers satisfying certain properties.

And while I believe your definition is equivalent to Dedekind's with some minor adjustments, it makes definition operations and proving theorems about the real numbers much harder.

Either way, reals are not defined as a limit or an infinite decimal representation.

John Gabriel

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Sep 9, 2017, 11:34:05 AM9/9/17
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I am right.

> We don't define pi in terms of pi.

How you define pi is not a problem. It is simply the ratio of the circumference to the diameter. The problem is that you call this ratio of magnitudes a **number** when you have no means of measuring it.

A number is the measure of a magnitude.

You can't get away from this. This definition is what it means to be a **number**.

> In the case of the Cauchy model, every real number is an equivalence class of Cauchy sequences.

Yes, but it is still nonsense because numbers are not defined in terms of elements but in terms of a chosen unit and measurement is not partial or approximate, it is COMPLETE. No sequence in an equivalence class is complete.

ALL the sequences in an equivalence class are related by the LIMIT which in the case if incommensurable magnitudes has not been proven to be a **NUMBER**. So it is very circular to state that an equivalence class defines a number, even if one were to accept sets as a valid representation, because the limit that is common to any Cauchy class has not been established as a rational number or anything else. If it is not a rational number, then it is an incommensurable magnitude.

> An an equivalence class is a set. In the case of the Dedekind construction, a real number is a set of rational numbers satisfying certain properties.
>
> And while I believe your definition is equivalent to Dedekind's with some minor adjustments, it makes definition operations and proving theorems about the real numbers much harder.

It debunks any proofs about objects which don't exist, viz. "real" numbers.

>
> Either way, reals are not defined as a limit or an infinite decimal representation.

They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.

Markus Klyver

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Sep 9, 2017, 12:02:18 PM9/9/17
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No, you're still wrong. A real number is not defined as a limit or an infinite decimal representation. That is just wrong and false.

I feel I begin to repeat myself, but In the case of the Cauchy model every real number is an equivalence class of Cauchy sequences. It's a set; not a limit or any sort. The *intuitive picture* you can have is of course some form of limit, but formally it is not a limit in any way.

It's not in any way circular either: every real number is a set. A particular set, namely an equivalence class of rational Cauchy sequences. A real number **is** an equivalence class. A particular equivalence class **is** a real number. I can't stress this enough.

All the Cauchy sequences in a certain equivalence class is related by the equivalence relation defining the equivalence class. They are not related by some "limit" of any kind. They are related by the equivalence relation.


I have touched on your definition of number before. You haven't defined what a measure or what a magnitude is. You have only given me synonyms such as "size" and "extent". It's equally vague and nonsensical. And equally, defining pi as the ratio of the circumference to the diameter of a circle requires you to to what lengths of curves are, for example. It also requires a rigours definition of what a circle is.

John Gabriel

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Sep 9, 2017, 1:52:28 PM9/9/17
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Chuckle. You are a moron therefore you are wrong. I have explained to you the correct concepts and corrected you over and over again. You keep responding with irrelevant crap and "Nah Uh".

<nonsense>

Markus Klyver

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Sep 9, 2017, 4:10:06 PM9/9/17
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In what sense am I wrong? You are the one calling people morons and anti-Semitic slur. Glad you don't teach anyone anymore; you can't even be corrected when wrong.

konyberg

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Sep 9, 2017, 4:20:07 PM9/9/17
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This look like a sentence in logic. Can you give where it comes from?

KON

zelos...@outlook.com

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Sep 11, 2017, 2:08:37 AM9/11/17
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On Saturday, September 9, 2017 at 7:52:28 PM UTC+2, John Gabriel wrote:
> Chuckle. You are a moron therefore you are wrong. I have explained to you the correct concepts and corrected you over and over again. You keep responding with irrelevant crap and "Nah Uh".
>
> <nonsense>

Gabriel, that is a pure example of an ad hominem fallacy.

We have corrected you over and over. Like your complaint about cauchy sequences, it doesn't matter that there are infinitely many different sequences because, AS WE HAVE ALREADY EXPLAINED TO YOU, it is the equivalence classes we care about! They are the ones we define as the real numbers.

zelos...@outlook.com

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Sep 11, 2017, 2:15:16 AM9/11/17
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On Saturday, September 9, 2017 at 5:34:05 PM UTC+2, John Gabriel wrote:
> They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.

The question is not wether or not the limit is a number, it is wether or not the limit is in Q, but the limit of course exists in R as the sequence itself is part of the equivalence classes that makes up R

John Gabriel

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Sep 11, 2017, 7:52:16 AM9/11/17
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In every sense. You are an ignoramus, a dishonest and morally bankrupt piece of shit like most of your Jewish masters. You are one very pathetic lackey.
You don't fool anyone any longer. I have made sure of this.

In due time you will be cleaning toilets which is more in line with your work aptitude.


zelos...@outlook.com

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Sep 11, 2017, 9:28:14 AM9/11/17
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See? This is what I mean, we show you your idiocy and you go on antisemitic garbage.

Dan Christensen

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Sep 11, 2017, 12:52:22 PM9/11/17
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On Wednesday, September 6, 2017 at 2:25:11 AM UTC-4, John Gabriel wrote:
> Jackie caused a tremendous amount of attention to be focused on my New Calculus with his scorching critique. After I refuted him...

Your are delusional, Troll Boy. You have never refuted anyone. Your Wacky New Calclueless is a dead end and a complete waste of time. You cannot even determine the derivative of functions as simple as y=x. No, Troll Boy, it is NOT "undefined." What a moron!

In your goofy number system, you have banned all axioms. So, of course, you cannot derive in the even most elementary results of basic arithmetic, not even 2+2=4.


Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

Markus Klyver

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Sep 16, 2017, 3:54:10 AM9/16/17
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To prove that Cauchy sequences in ℝ converges in ℝ is not too trivial. You first have to introduce a metric for real numbers and then show that real Cauchy sequences really converge in ℝ. The statement that Cauchy sequences in ℚ also converges in ℝ is trivial since ℚ is trivially embedded into ℝ. You just make equivalence relations between rationals and rational Cauchy sequences, which should be fairly obvious how to do.

Markus Klyver

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Sep 16, 2017, 3:55:59 AM9/16/17
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Maybe you could explain to us why you think we are wrong, instead of adhering to slur and insults.

John Gabriel

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Sep 16, 2017, 6:38:03 AM9/16/17
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Idiot. The explanation has been given to you many times. Now you get only insult because that is all you understand. Mooroooon!!!!!

John Gabriel

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Sep 16, 2017, 6:39:05 AM9/16/17
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On Saturday, 16 September 2017 03:54:10 UTC-4, Markus Klyver wrote:
> Den måndag 11 september 2017 kl. 08:15:16 UTC+2 skrev zelos...@outlook.com:
> > On Saturday, September 9, 2017 at 5:34:05 PM UTC+2, John Gabriel wrote:
> > > They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.
> >
> > The question is not wether or not the limit is a number, it is wether or not the limit is in Q, but the limit of course exists in R as the sequence itself is part of the equivalence classes that makes up R
>
> To prove that Cauchy sequences in ℝ converges in ℝ is not too trivial. You first have to introduce a metric for real numbers

Imbecile. You first have to show that a valid construction of "real" number exists. There is none.

<parrot crap>

Markus Klyver

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Sep 16, 2017, 3:11:06 PM9/16/17
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The Cauchy model is a valid one, but you don't understand it.

John Gabriel

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Sep 16, 2017, 5:16:45 PM9/16/17
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Grrr. You are stupid. This is what discourages me from trying to educate you. Listen boy, if I don't understand, then nobody else understands. Did you get this? I am far more intelligent than anyone I have ever known. This is not Dunning Kruger, it is a FACT.

You would be wise to stop telling me that I don't understand. It is YOU who do not understand.

Markus Klyver

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Sep 16, 2017, 7:34:53 PM9/16/17
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Then I'm quite surprised I have to tell you about how the Cauchy construction works. You obviously didn't know what a Cauchy sequence were, or how real numbers were defined in that model. You wouldn't state something stupid about how irrationals were defined as limits or infinite decimal expansions if you actually knew how the Cauchy construction worked.

John Gabriel

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Sep 16, 2017, 9:47:16 PM9/16/17
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Oh Shut up moron. You are an imbecile who knows nothing. I studied real analysis before you were born you dimwit.

Markus Klyver

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Sep 17, 2017, 3:31:26 AM9/17/17
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Oh, that's why you say constant rational sequences are not Cauchy sequences?

John Gabriel

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Sep 17, 2017, 3:44:59 AM9/17/17
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The main attribute of any Cauchy sequence is that it ***converges***. Everything else is irrelevant nonsense. This convergence IMPLIES a LIMIT.

ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.

Markus Klyver

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Sep 17, 2017, 5:05:15 AM9/17/17
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No, that's not the definition of a Cauchy sequence. And obviously not Cauchy sequences converge in ℚ. That's why we want real numbers; we *want* all Cauchy sequences to converge.

John Gabriel

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Sep 17, 2017, 9:37:54 AM9/17/17
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I did not say that was the definition of a Cauchy sequence. I piss and shit on the definitions of mainstream mythmaticians which I know well.

Read what I wrote carefully:

ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.

> And obviously not Cauchy sequences converge in ℚ.

It is NOT relevant what a Cauchy sequence converges to, only that it converges. If it converges to Q, then we know the limit is a number. If not Q, then the limit is NOT a number. Stop repeating what you read in your silly textbooks. I know all that and have studied it many years ago. I have a better understanding than anyone else.

> That's why we want real numbers;

That's why you want chocolate? Chuckle. Fool!


> we *want* all Cauchy sequences to converge.

What you want is irrelevant. Only facts matter.

Markus Klyver

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Sep 17, 2017, 11:24:51 AM9/17/17
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But you obviously have no idea what the definition of what a Cauchy sequence even is. No, all Cauchy sequences in ℚ does not converge in ℚ. Hence, we construct real numbers as equivalence classes of Cauchy sequences. The equivalence relation used says nothing about limits; it only requires the component-wise difference to approach zero. And not every Cauchy sequence is equal under a such equivalence relation, as demonstrated by the example I gave you where a_k = 1 for all k and b_k = 2 for all k.

If you are going to criticize real analysis, make sure you actually understand it first.

John Gabriel

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Sep 18, 2017, 8:53:37 AM9/18/17
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Hey moron, let me give you an example (hope you know what this is). To say the componentwise difference proves the sequences are equivalent is the SAME as saying their LIMIT is the same. The componentwise difference is a result of the LIMIT being the same.

The two series:

1/2 + 1/4 + 1/8 + ... = 1

1/3 + 1/9 + 1/27 + ... = 1/2

generate the sequences:

{1/2; 3/4; 7/8; ...}

{1/3; 4/9; 13/27; ...}

The componentwise difference of these sequences gives:

{ 1/6; 11/36; 85/216; ...}

And the limit of that last sequence is 1/2.

Do you get it now moron?

John Gabriel

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Sep 18, 2017, 9:01:52 AM9/18/17
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Your example of the sequences {1; 1; 1; ...} and {2; 2; 2; ...} prove that the LIMIT is the reason sequences are part of the same equivalence class. But you are too stupid to realise this.

zelos...@gmail.com

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Sep 18, 2017, 9:28:22 AM9/18/17
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Not at all, convergence does not imply a limit because the limit may not exist. That is why there are ways to construct expanded structures that allows the limit to exist.

zelos...@gmail.com

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Sep 18, 2017, 9:31:14 AM9/18/17
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Den lördag 16 september 2017 kl. 12:39:05 UTC+2 skrev John Gabriel:
There are plenty, you can show that Cauchy sequences, in the general sense that is not dependent on rational numbers, is valid,

Dedekind–MacNeille completion is another general construction method that is valid, on rational numbers you get the dedekind completion type.

You got the Eudoxus construction, which is perfectly valid as well.

There are many many more, all work and are generalized such that you can prove that they gives the desired results and for real numbers, are equivalent.

zelos...@gmail.com

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Sep 18, 2017, 9:32:39 AM9/18/17
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There is nothing that says you have to understand it for anyone else to understand it. That logical implication arrow does not exist. In reality most people in mathematics understand all of this vastely better than you. It is just your narcissistic crank ego that doesn't want it to be so.

And yes, it is Dunning-krüger, very much so, as we can clearly see you do not understand it at all.

zelos...@gmail.com

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Sep 18, 2017, 9:35:16 AM9/18/17
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No it doesn't, this does however demonstrate you have no clue what you are on about.

The sequences are distinct because their componentwise differens is not a null sequence. That is why, not about a limit.

John Gabriel

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Sep 18, 2017, 10:22:21 AM9/18/17
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Like almost everything else you are ignorant of the very subject you claim to know - real analysis.

It is a FACT that:

John Gabriel

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Sep 18, 2017, 10:24:29 AM9/18/17
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We are not talking about distinct sequences you ape. That is a GIVEN. If the component difference sequence is a null sequence then the sequences have the same LIMIT.

The example shows this my stupid one.

John Gabriel

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Sep 18, 2017, 11:35:11 AM9/18/17
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On Monday, 18 September 2017 08:53:37 UTC-4, John Gabriel wrote:
To prove me wrong, you would need to find a pair of sequences whose componentwise difference sequence is NULL and has NO limit.

Good luck moron!

Python

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Sep 18, 2017, 11:48:01 AM9/18/17
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John Gabriel wrote:
> To prove me wrong, you would need to find a pair of sequences whose
> componentwise difference sequence is NULL and has NO limit.

x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2

y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2



Markus Klyver

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Sep 18, 2017, 12:41:12 PM9/18/17
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It's not equivalent. If two rational Cauchy sequences have the same rational limit, then their component-wise difference approaches zero as n --> infinity. But the converse does not have to be true since a rational limit may not exist.

Also, by your "definition"

1/2 + 1/4 + 1/8 + ... = 1

1/3 + 1/9 + 1/27 + ... = 1/2

are both false because no finite sum will ever equal to the RHS.

Markus Klyver

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Sep 18, 2017, 12:43:24 PM9/18/17
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No, two Cauchy sequences are in the same equivalence class because they are equivalent under a equivalence relation. The equivalence relation we use when defining real numbers mentions nothing about limits.

Markus Klyver

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Sep 18, 2017, 12:44:54 PM9/18/17
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Again, this is false. All rational Cauchy sequences does not have a rational limit. That's we we want to extend the rationals to the reals.

Markus Klyver

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Sep 18, 2017, 12:47:59 PM9/18/17
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Sure. Take any sequences {a_k} and {b_k} which are not the same such that (a_n)^2 and (b_b)^2 both converges to 2 as n ---> infinity.

burs...@gmail.com

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Sep 18, 2017, 2:49:49 PM9/18/17
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I was first reading the Exodus construction.
But the Eudoxus construction is worth reading.

The Eudoxus Real Numbers
R. D. Arthan
(Submitted on 24 May 2004)
https://arxiv.org/abs/math/0405454

This one is also nice:

chapter 2 of John Harrison’s thesis [5];
John Harrison. Theorem Proving with
the Real Numbers. Technical report,
University of Cambridge Computer Laboratory, 1996.

Basically you can do Cauchy with integers only.

Markus Klyver

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Sep 18, 2017, 3:54:34 PM9/18/17
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Never seen a such construction before. Interesting.

John Gabriel

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Sep 18, 2017, 6:49:15 PM9/18/17
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They both have the limit sqrt(2). You haven't proven me wrong.

John Gabriel

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Sep 18, 2017, 6:51:31 PM9/18/17
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Do you suffer from Down Syndrome? The statement:

ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.

is true you moron!!!!

I did not say what kind of limit, only that for every Cauchy sequence, there is a limit. Nowhere did I say the limit is rational or not rational.

Learn to read dumbo!!!! This is the third time you make the same mistake!!!!

John Gabriel

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Sep 18, 2017, 6:53:45 PM9/18/17
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Every Cauchy sequence converges and every convergent sequence is a Cauchy sequence but the imbeciles Klyver and Zelos do not understand this. Chuckle.

http://www.maths.qmul.ac.uk/~ig/MAS111/Cauchy%20Criterion.pdf

John Gabriel

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Sep 18, 2017, 7:06:22 PM9/18/17
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Has ZERO to do with Eudoxus.

Python

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Sep 19, 2017, 5:00:08 AM9/19/17
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They both have no limit in the field of rationnal numbers Q.


John Gabriel

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Sep 19, 2017, 7:53:17 AM9/19/17
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On Tuesday, 19 September 2017 05:00:08 UTC-4, Python wrote:
> John Gabriel wrote:
> > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> >> John Gabriel wrote:
> >>> To prove me wrong, you would need to find a pair of sequences whose
> >>> componentwise difference sequence is NULL and has NO limit.
> >>
> >> x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> >>
> >> y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> >
> > They both have the limit sqrt(2). You haven't proven me wrong.
>
> They both have no limit in the field of rationnal numbers Q.

I didn't say they have a limit in the rational numbers. I said they have a "limit".

You should pay attention to detail. You have been so brainwashed by your "schooling", that you automatically see things that aren't there.

Of course there is no limit in the rational numbers which means that the limit is an incommensurable magnitude, NOT a real number.

Python

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Sep 19, 2017, 9:44:56 AM9/19/17
to
Your inability to get the point about Cauchy's construction of real
numbers is abysmal Mr Gabriel. You'd better find another hobby, math
is definitely not your cup of tea.



John Gabriel

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Sep 19, 2017, 2:26:41 PM9/19/17
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On Tuesday, 19 September 2017 09:44:56 UTC-4, Python wrote:
> John Gabriel wrote:
> > On Tuesday, 19 September 2017 05:00:08 UTC-4, Python wrote:
> >> John Gabriel wrote:
> >>> On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> >>>> John Gabriel wrote:
> >>>>> To prove me wrong, you would need to find a pair of sequences whose
> >>>>> componentwise difference sequence is NULL and has NO limit.
> >>>>
> >>>> x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> >>>>
> >>>> y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> >>>
> >>> They both have the limit sqrt(2). You haven't proven me wrong.
> >>
> >> They both have no limit in the field of rationnal numbers Q.
> >
> > I didn't say they have a limit in the rational numbers. I said they have a "limit".
> >
> > You should pay attention to detail. You have been so brainwashed by your "schooling", that you automatically see things that aren't there.
> >
> > Of course there is no limit in the rational numbers which means that the limit is an incommensurable > magnitude, NOT a real number.
>
> Your inability to get the point about Cauchy's construction of real
> numbers is abysmal Mr Gabriel.

So you finally admit you are wrong. That is a first step. An improvement!! Chuckle.

Cauchy's poop is abysmal and it swivels in your thinking processes. The only way to survive is to expel the poop so you can think clearly.

> You'd better find another hobby, math is definitely not your cup of tea.

Heh. Heh. That statement screws you more than it does me. Of course you are too ignorant to realise it. Chuckle.

Markus Klyver

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Sep 20, 2017, 4:37:45 PM9/20/17
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They don't have a rational limit, and it makes no sense to talk about convergence to a number we haven't established yet. Without real numbers, a such sequence will diverge.

Markus Klyver

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Sep 20, 2017, 4:42:50 PM9/20/17
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Den tisdag 19 september 2017 kl. 00:53:45 UTC+2 skrev John Gabriel:
> On Monday, 18 September 2017 18:51:31 UTC-4, John Gabriel wrote:
> Every Cauchy sequence converges and every convergent sequence is a Cauchy sequence but the imbeciles Klyver and Zelos do not understand this. Chuckle.
>
> http://www.maths.qmul.ac.uk/~ig/MAS111/Cauchy%20Criterion.pdf


You don't seem to understand. You cannot talk about a limit which doesn't exist. A rational sequence cannot converge to anything in ℚ and if an element to converge to doesn't exist, then the sequence is simply divergent.

Bolzano-Weierstrass' is about Cauchy sequences in ℝ, which is a complete field as I have mentioned. Every Cauchy sequence in ℝ converges in ℝ. This (usually) require real numbers to establish though.

burs...@gmail.com

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Sep 20, 2017, 5:10:16 PM9/20/17
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Maybe this could be authorative for what a tangent is:

Am Mittwoch, 20. September 2017 23:02:12 UTC+2 schrieb burs...@gmail.com:
> Ok, a prominent name and geometry:
>
> Hilbert, David; Cohn-Vossen, Stephan (1952).
> Geometry and the Imagination (2nd ed.). Chelsea.
> http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN379425343

Check out: §26 Ebene Kurven,
Abb. 182 and Abb. 183
(Page 155 and Page 156)

John Gabriel

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Sep 20, 2017, 7:30:49 PM9/20/17
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On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > John Gabriel wrote:
> > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > componentwise difference sequence is NULL and has NO limit.
> > >
> > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > >
> > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> >
> > They both have the limit sqrt(2). You haven't proven me wrong.
>
> They don't have a rational limit,

Correct moron. They don't and I never claimed they did.

> and it makes no sense to talk about convergence to a number we haven't established yet.

And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.

Pathetic. Unbelievably stupid moron you are.

> Without real numbers, a such sequence will diverge.

Chuckle.

So, according to you

{3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.

No my little orangutan, it converges on the incommensurable magnitude called PI. And yes, PI is not a number for otherwise my brilliant ancestors would never have called it an incommensurable magnitude.




Markus Klyver

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Sep 21, 2017, 7:53:22 AM9/21/17
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Except we don't. We define ℝ as the set of all equivalence classes of rational Cauchy sequences under the equivalence relationship

a ~ b <----> |a_n - b_n| approaches 0 as n approaches infinity


This is the exact construction of real numbers in the Cauchy model. We don't define real numbers as limits of infinite decimal expansions. We define ℝ as a set of equivalence classes.


Yes, the sequence containing the decimal expansion of pi diverges in ℚ. There simply isn't a rational number to converge towards. You need real numbers to claim every rational Cauchy sequence converges.

Unless you have a non-standard definition of a limit, rational sequences "approaching" irrational numbers is simply divergent in ℚ. It makes absolutely no sense under the standard limit definition to talk about "converging" Cauchy sequences that don't converge to an element in ℚ.

John Gabriel

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Sep 21, 2017, 7:59:04 AM9/21/17
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On Wednesday, 20 September 2017 19:30:49 UTC-4, John Gabriel wrote:
> On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> > Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > > John Gabriel wrote:
> > > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > > componentwise difference sequence is NULL and has NO limit.
> > > >
> > > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > > >
> > > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> > >
> > > They both have the limit sqrt(2). You haven't proven me wrong.
> >
> > They don't have a rational limit,
>
> Correct moron. They don't and I never claimed they did.
>
> > and it makes no sense to talk about convergence to a number we haven't established yet.
>
> And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.
>
> Pathetic. Unbelievably stupid moron you are.
>
> > Without real numbers, a such sequence will diverge.
>
> Chuckle.
>
> So, according to you
>
> {3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.

Can't you see how circular is the Cauchy "definition"? You claim to have real number when you don't have a rational number limit. In other words you assume the prior establishment of the LIMIT as a real number. But being of such low IQ as you are, you didn't notice.

Carl Boyer who was much smarter than you will ever be had this to say in his famous book on the history of calculus:

"Cauchy had stated in his Cours d'analyse that irrational numbers are to be regarded as the limits of sequences of rational numbers. Since a limit is defined as a number to which the terms of the sequence approach in such a way that ultimately the difference between this number and the terms of the sequence can be made less than any given number, the existence of the irrational number depends, in the definition of limit, upon the known existence, and hence the prior definition, of the very quantity whose definition is being attempted.


That is, one cannot define the number sqrt(2) as the limit of the sequence 1, 1.4, 1.41, 1.414, ... because to prove that this sequence has a limit one must assume, in view of the definitions of limits and convergence, the existence of this number as previously demonstrated or defined. Cauchy appears not to have noticed the circularity of the reasoning in this connection, but tacitly assumed that every sequence converging within itself has a limit."

The History of Calculus and its Conceptual Development' (Page. 281) Carl B. Boyer

John Gabriel

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Sep 21, 2017, 8:07:15 AM9/21/17
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On Thursday, 21 September 2017 07:53:22 UTC-4, Markus Klyver wrote:
> Den torsdag 21 september 2017 kl. 01:30:49 UTC+2 skrev John Gabriel:
> > On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> > > Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > > > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > > > John Gabriel wrote:
> > > > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > > > componentwise difference sequence is NULL and has NO limit.
> > > > >
> > > > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > > > >
> > > > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> > > >
> > > > They both have the limit sqrt(2). You haven't proven me wrong.
> > >
> > > They don't have a rational limit,
> >
> > Correct moron. They don't and I never claimed they did.
> >
> > > and it makes no sense to talk about convergence to a number we haven't established yet.
> >
> > And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.
> >
> > Pathetic. Unbelievably stupid moron you are.
> >
> > > Without real numbers, a such sequence will diverge.
> >
> > Chuckle.
> >
> > So, according to you
> >
> > {3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.
> >
> > No my little orangutan, it converges on the incommensurable magnitude called PI. And yes, PI is not a number for otherwise my brilliant ancestors would never have called it an incommensurable magnitude.
>
> Except we don't.

Except you DO!

> We define ℝ as the set of all equivalence classes of rational Cauchy sequences under the equivalence relationship

Yes, a very circular ill-formed and therefore BAD definition. The rational numbers can be EXPRESSED EXACTLY in terms of the chosen UNIT.

Your IMAGINARY real number CANNOT be expressed exactly for many reasons. The very first being you need the ellipsis which you claim is the LIMIT that you haven't established as a real number.

A number is NOT a set. In fact a sequence of numbers DOES NOT exist without prior establishment of NUMBER.

>
> a ~ b <----> |a_n - b_n| approaches 0 as n approaches infinity

Non-remarkable. It's a consequence of the fact that a series converges. Also NOTHING to do with infinity which is not even relevant.

<irrelevant crap>

Markus Klyver

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Sep 21, 2017, 8:20:14 AM9/21/17
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No, we don't define real numbers in terms of limits; that would be circular and nonsensical to do. Because the limit definition requires the limit to exist.

Just like you, Carl Boyer is wrong. We can axiomatically declare all rational Cauchy sequences to have a limit and axiomatically "construct" real numbers, but the Cauchy construction doesn't do that. We don't axiomatically declare real numbers exist; we define a real number as a particular equivalence class of rational Cauchy sequences.

WE don't define sqrt(2) in terms of limits. Using Cauchy's construction, take any Cauchy sequence {a_k} such that (a_n)^2 approaches 2 as n approaches infinity. Then we define sqrt(2) as the equivalence class [a].

[a] is an equivalence class, and not a limit of a particular Cauchy sequence. You fail to understand this very simple concept over and over again and you keep claiming things that are not true over and over again.

We do not define sqrt(2) as a limit of a rational sequence. A sequence can be constant, and all rational sequences are not convergent in ℚ.

Markus Klyver

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Sep 21, 2017, 8:29:34 AM9/21/17
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Except we don't. We define ℝ as the set of all equivalence classes of rational Cauchy sequences under a particular equivalence relationship. Equivalence classes are not limits. You can't even meaningfully talk about limits towards anything we haven't constructed yet. We use Cauchy sequences of rational numbers to construct reals, because we know how rational sequences work. It's not circular; you simply fail to understand the construction.

a ~ b <----> |a_n - b_n| approaches 0 as n approaches infinity
is a definition of what if means for two rational Cauchy sequences to be equal. It's not a theorem, nor a consequence. It's a definition. And as I said before, there are divergent Cauchy sequences in ℚ.

zelos...@gmail.com

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Sep 21, 2017, 8:33:01 AM9/21/17
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I am quite well versed in it unlike you. Again you fail to understand basic things.

A limit exists only if the element itself exist and most cauchy sequences of rational numbers, this is not the case. There is no rational number it converges to so there is no limit to speak of.

If however working within the real numbers, the limit always exists. Before you start complaining about others maybe you should learn to actually state your positions in clear detailed correct manners, rather than your asinine ways that is juvenile at best?

zelos...@gmail.com

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Sep 21, 2017, 8:36:18 AM9/21/17
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Den måndag 18 september 2017 kl. 16:24:29 UTC+2 skrev John Gabriel:
> On Monday, 18 September 2017 09:35:16 UTC-4, zelos...@gmail.com wrote:
> > Den måndag 18 september 2017 kl. 15:01:52 UTC+2 skrev John Gabriel:
> > > On Monday, 18 September 2017 08:53:37 UTC-4, John Gabriel wrote:
> > > > On Sunday, 17 September 2017 11:24:51 UTC-4, Markus Klyver wrote:
> > > > > Den söndag 17 september 2017 kl. 15:37:54 UTC+2 skrev John Gabriel:
> > > > > > On Sunday, 17 September 2017 05:05:15 UTC-4, Markus Klyver wrote:
> > > > > > > Den söndag 17 september 2017 kl. 09:44:59 UTC+2 skrev John Gabriel:
> > > > > > > > On Sunday, 17 September 2017 03:31:26 UTC-4, Markus Klyver wrote:
> > > > > > > > > Den söndag 17 september 2017 kl. 03:47:16 UTC+2 skrev John Gabriel:
> > > > > > > > > > On Saturday, 16 September 2017 19:34:53 UTC-4, Markus Klyver wrote:
> > > > > > > > > > > Den lördag 16 september 2017 kl. 23:16:45 UTC+2 skrev John Gabriel:
> > > > > > > > > > > > On Saturday, 16 September 2017 15:11:06 UTC-4, Markus Klyver wrote:
> > > > > > > > > > > > > Den lördag 16 september 2017 kl. 12:39:05 UTC+2 skrev John Gabriel:
> > > > > > > > > > > > > > On Saturday, 16 September 2017 03:54:10 UTC-4, Markus Klyver wrote:
> > > > > > > > > > > > > > > Den måndag 11 september 2017 kl. 08:15:16 UTC+2 skrev zelos...@outlook.com:
> > > > > > > > > > > > > > > > On Saturday, September 9, 2017 at 5:34:05 PM UTC+2, John Gabriel wrote:
> > > > > > > > > > > > > > > > > They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.
> > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > The question is not wether or not the limit is a number, it is wether or not the limit is in Q, but the limit of course exists in R as the sequence itself is part of the equivalence classes that makes up R
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > To prove that Cauchy sequences in ℝ converges in ℝ is not too trivial. You first have to introduce a metric for real numbers
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > Imbecile. You first have to show that a valid construction of "real" number exists. There is none.
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > <parrot crap>
> > > > > > > > > > > > >
> > > > > > > > > > > > > The Cauchy model is a valid one, but you don't understand it.
> > > > > > > > > > > >
> > > > > > > > > > > > Grrr. You are stupid. This is what discourages me from trying to educate you. Listen boy, if I don't understand, then nobody else understands. Did you get this? I am far more intelligent than anyone I have ever known. This is not Dunning Kruger, it is a FACT.
> > > > > > > > > > > >
> > > > > > > > > > > > You would be wise to stop telling me that I don't understand. It is YOU who do not understand.
> > > > > > > > > > >
> > > > > > > > > > > Then I'm quite surprised I have to tell you about how the Cauchy construction works. You obviously didn't know what a Cauchy sequence were, or how real numbers were defined in that model. You wouldn't state something stupid about how irrationals were defined as limits or infinite decimal expansions if you actually knew how the Cauchy construction worked.
> > > > > > > > > >
> > > > > > > > > > Oh Shut up moron. You are an imbecile who knows nothing. I studied real analysis before you were born you dimwit.
> > > > > > > > >
> > > > > > > > > Oh, that's why you say constant rational sequences are not Cauchy sequences?
> > > > > > > >
> > > > > > > > The main attribute of any Cauchy sequence is that it ***converges***. Everything else is irrelevant nonsense. This convergence IMPLIES a LIMIT.
> > > > > > > >
> > > > > > > > ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.
> > > > > > >
> > > > > > > No, that's not the definition of a Cauchy sequence.
> > > > > >
> > > > > > I did not say that was the definition of a Cauchy sequence. I piss and shit on the definitions of mainstream mythmaticians which I know well.
> > > > > >
> > > > > > Read what I wrote carefully:
> > > > > >
> > > > > > ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.
> > > > > >
> > > > > > > And obviously not Cauchy sequences converge in ℚ.
> > > > > >
> > > > > > It is NOT relevant what a Cauchy sequence converges to, only that it converges. If it converges to Q, then we know the limit is a number. If not Q, then the limit is NOT a number. Stop repeating what you read in your silly textbooks. I know all that and have studied it many years ago. I have a better understanding than anyone else.
> > > > > >
> > > > > > > That's why we want real numbers;
> > > > > >
> > > > > > That's why you want chocolate? Chuckle. Fool!
> > > > > >
> > > > > >
> > > > > > > we *want* all Cauchy sequences to converge.
> > > > > >
> > > > > > What you want is irrelevant. Only facts matter.
> > > > >
> > > > > But you obviously have no idea what the definition of what a Cauchy sequence even is. No, all Cauchy sequences in ℚ does not converge in ℚ. Hence, we construct real numbers as equivalence classes of Cauchy sequences. The equivalence relation used says nothing about limits; it only requires the component-wise difference to approach zero. And not every Cauchy sequence is equal under a such equivalence relation, as demonstrated by the example I gave you where a_k = 1 for all k and b_k = 2 for all k.
> > > > >
> > > > > If you are going to criticize real analysis, make sure you actually understand it first.
> > > >
> > > > Hey moron, let me give you an example (hope you know what this is). To say the componentwise difference proves the sequences are equivalent is the SAME as saying their LIMIT is the same. The componentwise difference is a result of the LIMIT being the same.
> > > >
> > > > The two series:
> > > >
> > > > 1/2 + 1/4 + 1/8 + ... = 1
> > > >
> > > > 1/3 + 1/9 + 1/27 + ... = 1/2
> > > >
> > > > generate the sequences:
> > > >
> > > > {1/2; 3/4; 7/8; ...}
> > > >
> > > > {1/3; 4/9; 13/27; ...}
> > > >
> > > > The componentwise difference of these sequences gives:
> > > >
> > > > { 1/6; 11/36; 85/216; ...}
> > > >
> > > > And the limit of that last sequence is 1/2.
> > > >
> > > > Do you get it now moron?
> > >
> > > Your example of the sequences {1; 1; 1; ...} and {2; 2; 2; ...} prove that the LIMIT is the reason sequences are part of the same equivalence class. But you are too stupid to realise this.
> >
> > No it doesn't, this does however demonstrate you have no clue what you are on about.
> >
> > The sequences are distinct because their componentwise differens is not a null sequence. That is why, not about a limit.
>
> We are not talking about distinct sequences you ape. That is a GIVEN. If the component difference sequence is a null sequence then the sequences have the same LIMIT.
>
> The example shows this my stupid one.

Again, the limit may not exist within the given structure.

zelos...@gmail.com

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Sep 21, 2017, 8:38:20 AM9/21/17
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> To prove me wrong, you would need to find a pair of sequences whose componentwise difference sequence is NULL and has NO limit.
>
> Good luck moron!

For a sequence with no limit in Q, how about (1/1,2/1, 3/2, 5/3,...)
where each element is the quotient of successive fibbonaci numbers? Tell me what rational number it converges to.

If that rational number doesn't exist, it has no limit in Q.

zelos...@gmail.com

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Sep 21, 2017, 8:39:41 AM9/21/17
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A number is whatever we decide it is as a logical construct. As all in mathematics is just logical constructs we can do as we please on that department. We use sets, sequences, pairs of stuff, etc.

John Gabriel

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Sep 21, 2017, 12:16:30 PM9/21/17
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Markus Klyver

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Sep 21, 2017, 1:46:15 PM9/21/17
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I have tried to explain these things for Gabriel quite many times now, but he doesn't get it. A limit may exist in one stricture but not in an other. I think it's pretty simple and should be understandable. But apparently we can meaningfully talk about limits that doesn't exist.
Den torsdag 21 september 2017 kl. 14:39:41 UTC+2 skrev zelos...@gmail.com:
> Den torsdag 21 september 2017 kl. 14:07:15 UTC+2 skrev John Gabriel:
> > On Thursday, 21 September 2017 07:53:22 UTC-4, Markus Klyver wrote:
> > > Den torsdag 21 september 2017 kl. 01:30:49 UTC+2 skrev John Gabriel:
> > > > On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> > > > > Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > > > > > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > > > > > John Gabriel wrote:
> > > > > > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > > > > > componentwise difference sequence is NULL and has NO limit.
> > > > > > >
> > > > > > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > > > > > >
> > > > > > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> > > > > >
> > > > > > They both have the limit sqrt(2). You haven't proven me wrong.
> > > > >
> > > > > They don't have a rational limit,
> > > >
> > > > Correct moron. They don't and I never claimed they did.
> > > >
> > > > > and it makes no sense to talk about convergence to a number we haven't established yet.
> > > >
> > > > And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.
> > > >
> > > > Pathetic. Unbelievably stupid moron you are.
> > > >
> > > > > Without real numbers, a such sequence will diverge.
> > > >
> > > > Chuckle.
> > > >
> > > > So, according to you
> > > >
> > > > {3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.
> > > >
> > > > No my little orangutan, it converges on the incommensurable magnitude called PI. And yes, PI is not a number for otherwise my brilliant ancestors would never have called it an incommensurable magnitude.
> > >
> > > Except we don't.
> >
> > Except you DO!
> >
> > > We define ℝ as the set of all equivalence classes of rational Cauchy sequences under the equivalence relationship
> >
> > Yes, a very circular ill-formed and therefore BAD definition. The rational numbers can be EXPRESSED EXACTLY in terms of the chosen UNIT.
> >
> > Your IMAGINARY real number CANNOT be expressed exactly for many reasons. The very first being you need the ellipsis which you claim is the LIMIT that you haven't established as a real number.
> >
> > A number is NOT a set. In fact a sequence of numbers DOES NOT exist without prior establishment of NUMBER.
> >
> > >
> > > a ~ b <----> |a_n - b_n| approaches 0 as n approaches infinity
> >
> > Non-remarkable. It's a consequence of the fact that a series converges. Also NOTHING to do with infinity which is not even relevant.
> >
> > <irrelevant crap>
>
> A number is whatever we decide it is as a logical construct. As all in mathematics is just logical constructs we can do as we please on that department. We use sets, sequences, pairs of stuff, etc.

THANK YOU! Yes, there isn't really an unified definition of what a number is and have to obey; it depends on the context. Often we mean real or complex numbers, but when constructing these things we use tuples, sets and other already established concepts to create new objects.

John Gabriel

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Sep 21, 2017, 2:25:38 PM9/21/17
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Huh? I didn't write that you idiot. You did! Chuckle. Boy, you are getting more lost with every comment.

Markus Klyver

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Sep 21, 2017, 5:07:20 PM9/21/17
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Except that you have. You don't know what a sequence is, much less what a Cauchy sequence is or how real numbers are defined. I, and several others, have tried to tell you why you are wrong but you keep adhering to racial slur and anti-Semitic crap. You are a 54 year old full-grown man with nothing better to do than bullshitting on mathematics you don't understand. Mathematics any undergraduate understands pretty well. You keep re-defining terminology and inventing your own. Then you call everyone retards for not getting "how smart you are" for misusing mathematical notation and using circular definitions.

John Gabriel

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Sep 21, 2017, 8:54:27 PM9/21/17
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Look tosser, every time you claim I stated anything, others can read my comments and quickly realise you are a delusional twerp.

> You don't know what a sequence is, much less what a Cauchy sequence is or how real numbers are defined.

It is YOU who do not understand what is a series or a sequence because you are an incorrigible fucking moron who will not listen to someone who is infinitely smarter than him and all the orangutans in mainstream media.

I did not discover the New Calculus because you are smart you moron! I discovered it because I am the smartest. Ask yourself why no one else was able to realise it before me. It won't help actually because you are a confirmed idiot.

<bullshit>

John Gabriel

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Sep 21, 2017, 8:57:33 PM9/21/17
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Walter Idiot Rudin (Page 59):

With {a_n} we associate a sequence {s_n} where s_n = \sum_{k=1}^n a_k.

This means that a sequence is derived from a series. Get it moron?

All this time, both you and Jan Bielawski have been claiming otherwise with Jan recognising you are wrong in the one attempt, but both of you do not understand what is a series or sequence.

You are a pathetic imbecile who knows shit about the very topic you are trying to defend.

John Gabriel

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Sep 21, 2017, 9:25:28 PM9/21/17
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That definition by Rudin also contradicts yours and Jan Bielawaki's claims that a sequence can be derived without a series.

But this is what I have come to expect from nincompoops like you.

zelos...@gmail.com

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Sep 22, 2017, 5:05:44 AM9/22/17
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You are the one that don't understand anything which has been shown time and time again and you as obsessed with your own narrow garbage rather than understand things on a much broader and wider scale. You are not the smartest, you are if anything, among the dumbest pieces of shit in mathematics ever.

zelos...@gmail.com

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Sep 22, 2017, 5:06:59 AM9/22/17
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No you fucking moron, it means that a series is derived from the concept of sequences you idiot.

Markus Klyver

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Sep 22, 2017, 6:27:54 AM9/22/17
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I know quite well what a sequence and an infinite series is, and I have given you formal definitions over and over again. You refuse to listen or learn anything new.

Markus Klyver

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Sep 22, 2017, 6:30:29 AM9/22/17
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I honestly don't know what you mean this. Of course you can derive a series from a sequence and a sequence from a series. But we were talking about sequences of rational numbers and how we can define real numbers as equivalence classes of rational Cauchy sequences.

Again, we do not define real numbers as infinite sums. We define them as equivalence classes. You don't seem to get this very crucial part.

zelos...@gmail.com

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Sep 22, 2017, 6:33:26 AM9/22/17
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He doesn't get anything, even if his life were to depend on it.

Markus Klyver

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Sep 22, 2017, 6:42:39 AM9/22/17
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A sequence is just a sequence. A sequence which can be (and is) the partial sums of a series. This because we define the value of an infinite series to be the limit of a particular sequence. But we often consider sequences without involving series at all.

adrianf...@gmail.com

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Sep 22, 2017, 7:00:48 AM9/22/17
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Hey John Gabriel, your BIG STUPID is showing.

John Gabriel

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Sep 22, 2017, 9:06:44 AM9/22/17
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Because you don't understand the topic you are trying to defend - real analysis. You have a very low reading comprehension IQ. You are confused by what I have told you because it is the FIRST time this has been pointed out to you whereas all you have heard are platitudes and handwaving in your courses.

> Of course you can derive a series from a sequence and a sequence from a series.

NO! Wrong again. Rudin defines a sequence as being derived from a series. You start with a series. You CANNOT have a sequence that is NOT derived from a series you IMBECILE! When will you understand this?

That's what is meant by:

----------------------------------------------------
Walter Idiot Rudin (Page 59):

With {a_n} we associate a sequence {s_n} where s_n = \sum_{k=1}^n a_k.
----------------------------------------------------

> But we were talking about sequences of rational numbers and how we can define real numbers as equivalence classes of rational Cauchy sequences.
>
> Again, we do not define real numbers as infinite sums.

You DO you moron, you DO! When you claim the infinite decimal digits define numbers, what are you doing? Tsk, tsk.

You define:

1.000...

and

0.999...

as equivalent Cauchy Sequences. They are equivalent because they hve the same limit!


> We define them as equivalence classes. You don't seem to get this very crucial part.

You do not get it.

Markus Klyver

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Sep 22, 2017, 4:01:27 PM9/22/17
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Now this is false. Rudin defines a sequence in Definition 2.7 in which a sequence is a function f defined from the naturals to some set J. He then denotes f(n) by x_n and the function f as a whole by {x_n}, all in accordance with standard notation.

And again, a sequence does not have to be "derived" from a series, even though you always can construct a series from any given sequence {s_n} such that its partial sums form the sequence {s_n}.- The vice versa is also true, you can always construct a sequence from the partial sums of a series.

We are not defining real numbers as infinite decimal expansions; I have repeatedly told you that. 1.000... is a real number, which is not a sequence. In the Cauchy model 1.000... is *an equivalence class* of rational Cauchy sequences. 0.999... is the same equivalence class.

1.000... is a notation for a real number, and in this notation 0.999... = 1.000... This notation is defined using limits.

Neither of them are sequences, and your misuse of mathematical language exposes you actually have no knowledge about this. We define reals as equivalence classes. You don't get this very crucial part, and keep insisting the Cauchy model is circular when it isn't.

John Gabriel

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Sep 22, 2017, 8:49:03 PM9/22/17
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No. It is true. 100% true. Chuckle.

> Rudin defines a sequence in Definition 2.7 in which a sequence is a function f defined from the naturals to some set J.

Wrong yet again. He defines a sequence as the PARTIAL SUMS of a series. That the partial sums are given by a summation function is completely UNREMARKABLE.

> And again, a sequence does not have to be "derived" from a series,

No. It is clear from Rudin that his sequences are derived from a series.

> even though you always can construct a series from any given sequence {s_n} such that its partial sums form the sequence {s_n}.- The vice versa is also true, you can always construct a sequence from the partial sums of a series.
>
> We are not defining real numbers as infinite decimal expansions;

Yes, you are moron. You are.

1 = 1+0+0+ ... = 1.000...

0.999... = 0.9 + 0.09 + 0.009 + ...

> I have repeatedly told you that. 1.000... is a real number, which is not a sequence. In the Cauchy model 1.000... is *an equivalence class* of rational Cauchy sequences. 0.999... is the same equivalence class.

I have repeatedly told you that S = Lim S is a very ill-formed definition.
You are a moron.

>
> 1.000... is a notation for a real number,

No. 1 is a notation for the standard unit which is a RATIONAL NUMBER. There are no real numbers.

>and in this notation 0.999... = 1.000... This notation is defined using limits.

Wrong. S = Lim S is an ill-formed definition.

>
> Neither of them are sequences, and your misuse of mathematical language exposes you actually have no knowledge about this. We define reals as equivalence classes. You don't get this very crucial part, and keep insisting the Cauchy model is circular when it isn't.

The Cauchy model is circular because it is circular.

Markus Klyver

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Sep 23, 2017, 8:22:08 AM9/23/17
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No. Read Definition 2.7 again.


Definition 2.7 reads: "By a sequence, we mean a function f defined on a set J of all the positive integers. If f(n) = x_n [...] it is customary to denote the sequence by {x_n}"

And no, we are not defining real numbers as infinite decimal expansions. This is a lie you keep repeating. We do not define real numbers as infinite decimal expansions. Instead, they can be defined as equivalence classes of rational Cauchy sequences. Equivalence classes are not limits, nor infinite decimal strings. We define decimal expansions in terms of limits, but this require the limit to exist. The limit may or may not exist without reals, but every monotonic and bounded sequence in ℝ is guaranteed to have a limit in ℝ.

And as I said, 1.000... is a real number, which is not a sequence. In the Cauchy model 1.000... is *an equivalence class* of rational Cauchy sequences. 0.999... is the same equivalence class.

In fact, both the decimal expansions 1.000... and 0.999... is well-defined without real numbers, as the limit clearly exist in ℚ. So even without ℝ you would have 0.999... = 1.000... = 1.


You haven't demonstrated how the Cauchy construction is circular, much less you actually understand it at all.

John Gabriel

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Sep 23, 2017, 10:31:05 AM9/23/17
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Listen idiot, I don't give a crap about Rudin - he was an idiot. The only reason I referred to his definitions is to show that you did not understand them.

THERE IS NO DEFINITION 2.7 IN THE THIRD EDITION.

> Definition 2.7 reads: "By a sequence, we mean a function f defined on a set J of all the positive integers. If f(n) = x_n [...] it is customary to denote the sequence by {x_n}"
>
<irrelevant crap refuted many times over>

Markus Klyver

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Sep 23, 2017, 2:22:59 PM9/23/17
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There is a Definition 2.7 on page 26 in the third edition. I referred to Rudin because you would see that standard literature sure use the definitions I use.

John Gabriel

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Sep 23, 2017, 2:38:22 PM9/23/17
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It is no different to deriving a sequence from a series.

Consider the series: 3/10 + 3/100 + 3/1000 + ...

The general term is given by 3/10^n where n >=1 and n is an +ve integer.

You can take the function f(x) = 1/3 (1 - 10^(-x)) as the sum and from it determine the sequence:

f(0) = 0
f(1) = 0.3
f(2) = 0.33

Therefore,

{0; 0.3; 0.33; 0.333; ...}

And note that the original function f(x) = 3/10^x, is a continuous function.

Conclusion: Definition 2.7 does not contradict anything on page 59, nor is it in a different context at all.

But can a monkey like you understand this? Nah.

Markus Klyver

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Sep 23, 2017, 2:48:04 PM9/23/17
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Except that we can consider sequences without invoking the concept of series at all.

John Gabriel

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Sep 23, 2017, 5:37:43 PM9/23/17
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Except you really can't because all sequences are as I explained derived from series which are derived from functions being indexed at fixed intervals. But there is nothing remarkable there.

It is sufficient for us to say that sequences are derived from series. Nothing else is needed.

Markus Klyver

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Sep 24, 2017, 8:23:12 AM9/24/17
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You can always come up with a series that have a particular sequence as its partial sums, yes. But we do not define sequences from sequences. We define series from sequences.

And all this, we still don't define real numbers as limits of rational Cauchy sequences. We define ℝ as the set of all equivalence classes of rational Cauchy sequences under the equivalence relationship

John Gabriel

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Sep 24, 2017, 8:32:19 AM9/24/17
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You are such a fool. Rudin makes it clear that sequences are defined from series. Remember that's who you are trying to defend monkey!

Markus Klyver

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Sep 24, 2017, 8:38:02 AM9/24/17
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But Rudin makes no such definition. Read Definition 2.7 on page 26. https://i.imgur.com/aBN4eFl.png

John Gabriel

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Sep 24, 2017, 8:41:23 AM9/24/17
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Imbecile. Rudin makes exactly that definition. Both 2.7 and the definition on page 59 are the SAME thing. You are too stupid to realise this. Definition 2.7 does not contradict Page 59 you idiot!!!!!!!!!!!!!!!!!!!

Me

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Sep 24, 2017, 8:43:38 AM9/24/17
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On Sunday, September 24, 2017 at 2:32:19 PM UTC+2, John Gabriel wrote:

> Rudin makes it clear that sequences are defined from series.

Idiot.

Markus Klyver

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Sep 24, 2017, 8:55:44 AM9/24/17
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If you read Definition 2.7, it should be clear that a sequence is more primitive than a series, since the concept of the value of a series relies on the concept of a sequence.

genm...@gmail.com

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Sep 24, 2017, 9:38:05 AM9/24/17
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Bollocks. Just because Rudin decided to write it before the page 59 definition? They are the SAME you imbecile. They are EQUIVALENT. What language do you want me to tell you this? Only thing is they are stated in a different way. Is that too hard for you also? Chuckle

Markus Klyver

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Sep 25, 2017, 11:50:52 AM9/25/17
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No, you're wrong. Definition 2.7 defines what a sequence is. Page 59 defines what the value of a series is, in terms of sequences.

In fact, some textbooks defines a series as a sequence.

John Gabriel

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Sep 25, 2017, 2:53:20 PM9/25/17
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Wrong again idiot. It defines a sequence from a series. The definition is no different to page 59.

Your professor is an idiot and that explains why you are an idiot.

Markus Klyver

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Sep 25, 2017, 3:53:41 PM9/25/17
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It clearly states that the value of a series is considered to be limit of a particular sequence.

John Gabriel

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Sep 25, 2017, 6:37:08 PM9/25/17
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Exactly! Now you got it!

It clearly states that the value (Lim S) of a series (S) is considered to be limit of a particular sequence (Lim S).

Good boy!

Markus Klyver

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Sep 26, 2017, 4:06:25 PM9/26/17
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What do you mean by Lim S and S? S is already defined as a limit.

John Gabriel

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Sep 26, 2017, 6:36:10 PM9/26/17
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You are more dense than diamond. What a moron.
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