Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


Math Forum
»
Discussions
»
sci.math.*
»
sci.math
Notice: We are no longer accepting new posts, but the forums will continue to be readable.
Topic:
"Every Cauchy sequence of real numbers converges to a limit." BUT orangutan Klyver implies that Q is not part of the real numbers. Tsk, tsk.
Replies:
1
Last Post:
Sep 30, 2017 9:20 AM




Re: "Every Cauchy sequence of real numbers converges to a limit." BUT orangutan Klyver implies that Q is not part of the real numbers. Tsk, tsk.
Posted:
Sep 30, 2017 9:20 AM


Den fredag 29 september 2017 kl. 15:14:33 UTC+2 skrev John Gabriel: > On Friday, 29 September 2017 07:23:02 UTC5, Zelos Malum wrote: > > Den fredag 29 september 2017 kl. 14:10:56 UTC+2 skrev John Gabriel: > > > On Wednesday, 27 September 2017 15:53:06 UTC5, John Gabriel wrote: > > > > I hate the ignorance, arrogance, stupidity and hypocrisy of mainstream academics. Let me illustrate this by means of an example using the concept of a Cauchy sequence. > > > > > > > > "Every Cauchy sequence of real numbers converges to a limit." > > > > > > > > In spite of this, you will hear many mainstreamers talking about Q as if it is not part of R. > > > > > > > > For example, mainstreamers will harp on the irrelevant fact that a Cauchy sequence of rationals may not converge to a rational number, BUT this does not alter the fact that ALL Cauchy sequences converge. > > > > > > > > Thus, if every Cauchy sequence converges to some *LIMIT*, then the limit must be DEFINED in each case. Well, to mainstreamers the circularity of their definitions is not obvious. > > > > > > > > They will try to adorn their recognition of patterns in sequences. By stating an observation about Cauchy sequences whose main attribute is a LIMIT (one doesn't have convergence without a LIMIT. A limit is an "UPPER BOUND" or "LOWER BOUND") using symbols, they imagine themselves to be sophisticated or "formal". > > > > > > > > Chuckle. I suppose stating a definition without symbols is like attending a meeting in jeans and tshirt? > > > > > > > > An orangutan in a suit is an orangutan no less. > > > > > > > > "Informal" (aka English language version, chuckle) definition: > > > > > > > > A sequence of rational numbers (***) is called Cauchy, if for any random value, say ? and an index N into the sequence, the distance between any two consecutive terms whose indexes are both greater than N, is less than ?. > > > > > > > > (***) > > > > "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 > > > > > > > > "Formal" definition: > > > > > > > > A sequence of real numbers {a_n} is a Cauchy sequence provided that for every ? > 0, there is a natural number N so that when n, m ? N, we have that.  a_n ? a_m.  ? ?. > > > > > > > > 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 > > > > > > Almost every Tom, Dick and Moron gets overwhelmed and simply resorts to parroting what their moron lecturers brainwash into their skulls. > > > > At least they can parrot, you fail even at that. > > Of course. Because I am not a parrot. Chuckle. > > The article you tried to refute but saw your arse: > > https://drive.google.com/open?id=0BmOEooW03iLSTROakNyVXlQUEU
You can't cause you are too stupid to understand basic definitoins.



