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Topic:
Teaching moronic trolls like JG about their own flawed theories
Replies:
5
Last Post:
Oct 4, 2017 6:48 AM




Re: Teaching mainstream morons about their own flawed theories: Every Cauchy sequence of real numbers converges to a limit.
Posted:
Oct 3, 2017 12:13 PM


Den onsdag 27 september 2017 kl. 15:53:39 UTC+2 skrev John Gabriel: > "Every Cauchy sequence of real numbers converges to a limit." > > http://math.caltech.edu/~nets/lecture4.pdf > > In spite of this, you will get thousands of morons (Klyver and "Me" and Burse included) talking about Q as if it is not part of R. > > For example the brainwashed moron Klyver will harp on the irrelevant fact that a Cauchy sequence of rationals may not converge to a rational number, BUT this does not mean the sequence does not converge because ALL Cauchy sequences WITHOUT ANY EXCEPTIONS converge. > > Thus, if every Cauchy sequence converges to some *LIMIT*, then the limit must be DEFINED in each case. Well, to mainstream morons the circularity of their definitions is oblivious because syphilitic brains are unable to think properly. > > Carl Boyer summed it up best: > > "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 > > Orangutan academics will try to adorn their recognition of patters in series. By stating an observation about Cauchy sequences whose main attribute is a LIMIT (Because guess what you infinitely stupid imbeciles? You don't have convergence without a LIMIT. A limit in a Cauchy sequence is equivalent to the expression "UPPER BOUND" or "LOWER BOUND") using symbols, they imagine themselves to be sophisticated or "formal". > > Chuckle. I suppose that stating a definition without symbols is like attending a meeting in jeans and tshirt, whereas the meeting is attending in a suit when symbols are used. Only problem is that a monkey in a suit is a monkey no less. > > > The Informal 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 ?. > > Now for the "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.  ? ?. > > Moron Klyver pay attention! It says "sequence of real numbers", but there are never any other sequences besides those of "rational numbers". Did you get this you baboon? Chuckle. It is irrelevant whether a sequence converges in the rationals or the mythical "reals". > > 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. > > gilstrang@gmail.com (MIT) > huizenga@psu.edu (HARVARD) > andersk@mit.edu (MIT) > david.ullrich@math.okstate.edu (David Ullrich) > djoyce@clarku.edu > markcc@gmail.com
But again, this applies to real Cauchy sequence. A rational Cauchy sequence does not have to converge to a rational limit, which is exactly the point. So defining a real number as a limit of rational numbers is doomed from the very beginning, and every mainstream mathematician would agree. That's why we *don't* construct real numbers in a such way. It's fully possible that Cauchy stated a such thing, and it's also how we intuitively think about real numbers, but it's not how real numbers are defined or how Cauchy defined them. This, assuming Cauchy even defined real numbers which I'm not sure of. I'm not sure how by who or when the first rigorous construction of real numbers was made.
Consider rational Cauchy sequences, that is a sequence of rational numbers {a_n} such that for every ? > 0, there is a natural number N so that when n, m ? N, we have that.  a_n ? a_m.  ? ?. We now define a relation R on the set of all rational Cauchy sequences and say that two sequences {a_n} and {b_n} relates to each other if and only if the limit of a_n  b_n as n approaches infinity is zero. This relation R will be an equivalence relation and we may use the notation {a_n} ~ {b_n} for {a_n} R {b_n}. Now define a set {x : {a_n} ~ x}. This set is called an equivalence class for the sequence {a_n} under the equivalence relation ~. A such set is a real number.
To prove completeness of the set of all such equivalence classes, we first have to introduce a metric onto it. We can then, with this metric, formally prove the reals are complete. The metric we use is d(x, y) := xy where x and y now are real numbers. Subtraction of real numbers can be defined with componentwise subtraction of Cauchy sequences, and xy is then defined to be either xy or yx and we choose the nonnegative alternative. It actually doesn't matter what Cauchy sequence in a real number you choose to define operations on; they all result in an unique equivalence class. For example, we may consider the real numbers [{a_n}] and [{b_n}]. Now take an arbitrary Cauchy sequence {c_d} from [{a_n}] and an arbitrary Cauchy sequence {d_n} from [{b_n}]. Now [{c_d  d_n}] will be uniquely defined.
Den onsdag 27 september 2017 kl. 17:40:31 UTC+2 skrev John Gabriel: > On Wednesday, 27 September 2017 09:53:39 UTC4, John Gabriel wrote: > > "Every Cauchy sequence of real numbers converges to a limit." > > > > http://math.caltech.edu/~nets/lecture4.pdf > > > > In spite of this, you will get thousands of morons (Klyver and "Me" and Burse included) talking about Q as if it is not part of R. > > > > For example the brainwashed moron Klyver will harp on the irrelevant fact that a Cauchy sequence of rationals may not converge to a rational number, BUT this does not mean the sequence does not converge because ALL Cauchy sequences WITHOUT ANY EXCEPTIONS converge. > > > > Thus, if every Cauchy sequence converges to some *LIMIT*, then the limit must be DEFINED in each case. Well, to mainstream morons the circularity of their definitions is oblivious because syphilitic brains are unable to think properly. > > > > Carl Boyer summed it up best: > > > > "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 > > > > Orangutan academics will try to adorn their recognition of patters in series. By stating an observation about Cauchy sequences whose main attribute is a LIMIT (Because guess what you infinitely stupid imbeciles? You don't have convergence without a LIMIT. A limit in a Cauchy sequence is equivalent to the expression "UPPER BOUND" or "LOWER BOUND") using symbols, they imagine themselves to be sophisticated or "formal". > > > > Chuckle. I suppose that stating a definition without symbols is like attending a meeting in jeans and tshirt, whereas the meeting is attending in a suit when symbols are used. Only problem is that a monkey in a suit is a monkey no less. > > > > > > The Informal 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 ?. > > > > Now for the "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.  ? ?. > > > > Moron Klyver pay attention! It says "sequence of real numbers", but there are never any other sequences besides those of "rational numbers". Did you get this you baboon? Chuckle. It is irrelevant whether a sequence converges in the rationals or the mythical "reals". > > > > 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. > > > > gilstrang@gmail.com (MIT) > > huizenga@psu.edu (HARVARD) > > andersk@mit.edu (MIT) > > david.ullrich@math.okstate.edu (David Ullrich) > > djoyce@clarku.edu > > markcc@gmail.com > > There are many ways to state the definition of Cauchy sequence. > > Notice that irrational number is defined by Cauchy HIMSELF in terms of LIMITS, NOT EQUIVALENCE CLASSES which are a NONREMARKABLE CONSEQUENCE of limits. > > "Cauchy had stated in his Cours d'analyse that irrational numbers are to be regarded as the limits of sequences of rational numbers." > > The "Informal" definition of LIMIT: > > "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,..." > > Notice again, this limit is a number, a RATIONAL NUMBER, not an infinite sequence without any measure. A limit is not the EXACT measure of a convergent sequence, for otherwise it would not be called a limit, but rather the sequences ITSELF! Chuckle. > > Formally (***), the definition is given as: > > If for every epsilon greater than zero, there is a delta greater than zero, such that f(x)L<epsilon whenever 0<xc<delta, THEN L = Lim {x \to c}. > > > "...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." > > This circularity is immediately evident in the formal definition (***) of limit. For starters, to prove that a limit exists, one has to know it. This is not such an outrageous idea if it has nothing to do with calculus, but alas the DERIVATIVE is defined using this BULLSHIT!!! Chuckle. > > In order to calculate a derivative using "first orangutan principles", one MUST perform nonsense arithmetic and algebra. However, the derivative is stated as a LIMIT!!! This means that the LIMIT must be known before hand as I proved to Anders Kaesorg of MIT. > > The BIG STUPID cannot be turned.
Again, you don't understand the definition. The limit definition doesn't require L to be "known", whatever that means. We simply refer to a such L as a limit, and we can prove that given its existence it has to be unique. It is an other thing to prove or find a limit, but nothing in the definition requires us to do so.
Den torsdag 28 september 2017 kl. 01:00:48 UTC+2 skrev John Gabriel: > On Wednesday, 27 September 2017 18:49:26 UTC4, Me wrote: > > On Wednesday, September 27, 2017 at 3:53:39 PM UTC+2, John Gabriel wrote: > > > > > "Every Cauchy sequence of real numbers converges to a limit." > > > > > > http://math.caltech.edu/~nets/lecture4.pdf > > > > Yes, idiot. In the context of the reals, that is. *sigh* > > What you said moron? Are rational numbers not reals? Bwaa haaa haaa. You are a fine idiot! > > > Man, these lectures are dealing with real analysis. > > Of course idiot. Anyone who has studied real analysis and Rudin's shitty text know this. Your point moron? Ah, I see. You have none. > > Real analysis  that topic all about a nonexistent mythical object called "real number". Chuckle. > > > > In spite of this, you will get thousands of idiots (Klyver and "Me" and Burse included) talking about Q as if it is not part of R. > > > The reason for this is that if we "build up" the number systems, > > You build up shit because orangutans don't know shit about anything, never mind mathematics. > > > > For example [...] Klyver will harp on the [...] fact that a Cauchy sequence of rationals may not converge to a rational number > > > Right, that's exacty the relevant point. > > There is no relevance moron. This is why I correct you over and over again but your delusion prevents you from learning. > > > > [and] this [just] mean[s] the sequence does not converge > > Wrong. It does converge. The concept of convergence is not restricted to a particular class of a numbers you moron! It applies to any numbers. > > > > ... if every Cauchy sequence converges to some *LIMIT* ... > > > > In the context of the rational numbers it's NOT the case that every Cauchy sequence converges to some *LIMIT*. > > Of course it is moron. They converge to some LIMIT orangutans imagine to be a real number. Chuckle.
If you don't have real numbers, not every Cauchy sequence will converge no. The limit definition requires a metric to be defined between points and the limit, and certainly that L exists. You can't have a metric defined for elements that don't exist. It makes absolutely no sense to talk about a limit which doesn't exist in the structure you are considering.
Den torsdag 28 september 2017 kl. 01:08:54 UTC+2 skrev John Gabriel: > On Wednesday, 27 September 2017 09:53:39 UTC4, John Gabriel wrote: > > "Every Cauchy sequence of real numbers converges to a limit." > > > > http://math.caltech.edu/~nets/lecture4.pdf > > > > In spite of this, you will get thousands of morons (Klyver and "Me" and Burse included) talking about Q as if it is not part of R. > > > > For example the brainwashed moron Klyver will harp on the irrelevant fact that a Cauchy sequence of rationals may not converge to a rational number, BUT this does not mean the sequence does not converge because ALL Cauchy sequences WITHOUT ANY EXCEPTIONS converge. > > > > Thus, if every Cauchy sequence converges to some *LIMIT*, then the limit must be DEFINED in each case. Well, to mainstream morons the circularity of their definitions is oblivious because syphilitic brains are unable to think properly. > > > > Carl Boyer summed it up best: > > > > "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 > > > > Orangutan academics will try to adorn their recognition of patters in series. By stating an observation about Cauchy sequences whose main attribute is a LIMIT (Because guess what you infinitely stupid imbeciles? You don't have convergence without a LIMIT. A limit in a Cauchy sequence is equivalent to the expression "UPPER BOUND" or "LOWER BOUND") using symbols, they imagine themselves to be sophisticated or "formal". > > > > Chuckle. I suppose that stating a definition without symbols is like attending a meeting in jeans and tshirt, whereas the meeting is attending in a suit when symbols are used. Only problem is that a monkey in a suit is a monkey no less. > > > > > > The Informal 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 ?. > > > > Now for the "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.  ? ?. > > > > Moron Klyver pay attention! It says "sequence of real numbers", but there are never any other sequences besides those of "rational numbers". Did you get this you baboon? Chuckle. It is irrelevant whether a sequence converges in the rationals or the mythical "reals". > > > > 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. > > > > gilstrang@gmail.com (MIT) > > huizenga@psu.edu (HARVARD) > > andersk@mit.edu (MIT) > > david.ullrich@math.okstate.edu (David Ullrich) > > djoyce@clarku.edu > > markcc@gmail.com > > In order to get a quick introduction, I suggest you watch my one minute video. > > Euler made a mistake in defining S = Lim S. No matter how hard modern academics have tried to defend this illformed definition and to deny it most recently, the truth is hard to refute and stands out. > > Page 91 and 92 of Euler's Elements of Algebra is clear and irrefutable evidence that Euler defined S = Lim S. > > S = 1 + 1/2 + 1/4 + 1/8 + ... > > Lim S = 2 > > If we continue the series (S) to infinity, there will be no difference at all between its sum (infinite sum), and the value of the fraction 1/(1a) (* Lim S), or 2. > > It is just incredible how academics of the BIG STUPID (mainstream academia) have denied this consistently. The most recent denier was David Ullrich from OK state university. > > It is a fact that Euler defined S = Lim S. > > (*) The limit. > > ===================================================== > > Without much ado, here is the original German text with my translations and commentary following: > > 295. > > 1+1/2+1/4+... ohne ende. Denn nimmt man nur zwei elieber, so hat man 1+1/2, und so fehlet noch 1/2. Nimmt man drei elieber, so hat man 7/4, fehlet noch 1/4: nimmt man vier elieber, so hat man 15/8, fehlet noch 1/8: woraus man sieht, das immer weniger fehlet, folglich, wenn man unendlich weit fortgeht, so mus gar nichts fehlen. > > 1 + 1/2 + 1/4 + ... without end. If one takes two terms, then one has 3/2, and so 1/2 still remains. If one takes three terms, then one has 7/4 with 1/4 remaining. If one takes four terms, then one has 15/8 with 1/8 remaining, from which one observes that which remains is less each time. Consequently, if one proceeds infinitely, nothing remains. > > COMMENTARY: Notice that Euler believed that one could proceed infinitely, that is, *add up all the terms* into a 'infinite' sum. He truly believed that eventually nothing would remain. This was Euler's first reference to an infinite sum. > > 296. > > Man sehe a = 1/3, so wird unser Bruch 1/(1a) = 1/(11/3)=3/2, welchem daher folgende Reihe gleich ist 1+1/3+1/9+... bis ins unendliche. Nimmt man zwei elieber, so hat man 4/3, fehlet noch 1/6. Nimmt man vier elieber, so hat man 40/37, fehlet noch 1/54. Da nun der Fehler immer dreimal fleiner wird, so mus derselbe endlich versdwinden. > > If a = 1/3, then we have 1 / (1a) = 1 / (11 / 3) = 3/2, So the series for this fraction is equal to 1 + 1/3 + 1/9 + ... when taken to infinity. If one takes two terms, one has 4/3 with 1/6 remaining. If one takes four terms, one has 40/37 with 1/54 remaining. Since the error always decreases, it must eventually vanish. > > COMMENTARY: In 296, Euler elaborates further on the process taken to infinity with the same conclusion, that is, all the terms are summed and nothing remains  it all vanishes. > > It is very clear that Euler believed in an infinite sum. In today's mathematics, we know that 1 + 1/3 + 1/9 + ... has a limit of 3/2. But Euler wasn't happy just to think of it as an upper bound, he dogmatically stated that the sum is indeed equal to the limit. And he again states this candidly in 296. > > Euler calls his series (Reihe) 1 + 2/3 + 4/9 + ... (S) and the sum (or limit as we know it today Lim S) 3. Euler equates these two objects in 298, that is, S and Lim S. > > 298. > > Daher ist unser Bruch 1/(1+a) gleich dieser unendlichen Reihe: > > 1  a + aa  aaa + aaaa  ... > > Hence the fraction 1 / (1 + a) {Lim S} is equal to this infinite series: > > 1  a + aa  aaa + aaaa  ... {S} > > COMMENTARY: In 298, Euler leaves no doubt that S equals Lim S as he states this clearly, and not just with examples as one sees in the previous cases. > > And this is the evidence which cannot be refuted. Only a ignorant, dishonest and incompetent academic will still scream and shout. > > Euler's original text can be found here: > > http://reader.digitalesammlungen.de/... > > Conclusion: > > Some may even try to argue that the definition is well formed. This is quickly dismissed by the fact that infinity is a junk concept which can't be reified in any way whatsoever. > > Till this day we have fallacies such as 0.333... = 1/3. > > This fallacy is quickly dismissed by knowledge of the following number theorem: > > Given any fraction p/q, then an equivalent fraction can be found in base b, if and only if, all the prime factors of q, are also factors of b. > > On that theorem alone, it is mindboggling how anyone can claim 0.333... = 1/3. > > Let's move along and think about the limit of the series 0.3+0.03+0.003+... > > It can be written as follows: > > S = 3/10 + 3/100 + 3/1000 + ... + 3/10^n + 1/3 x (1/10^n) > > But the tail part, that is, 1/3 x (1/10^n) is chopped off and we are left with a supposedly "infinite" series: > > S = 3/10 + 3/100 + 3/1000 + ... + 3/10^n + ... > > S = 1/3 [1  10^(n) ] + ... > > Lim (n > oo) S = Lim (n > oo) 1/3 [1  10^(n) ] + ... = 3/10 / (1  1/10) = 1/3 > > As n > oo, it is clear that the right hand side above will have 1/3 as its limit. > > So according to Euler, S which is equal to 3/10 + 3/100 + 3/1000 + ... + 3/10^n + ... is also equal to Lim (n > oo) S which is equal to 1/3. That is, S = Lim S. > > It is very easy to see that the limit is 1/3. But to equate the series to its limit?! That's absolutely senseless. Academics might claim that at infinity, the sum will be 1/3, but as we've seen, the number theorem rejects that. Hence, it's not possible to represent 1/3 in base 10, as anything else besides a rational number approximation. > > The next retort is that 0.333... is only a symbol. Well, this is ridiculous because 1/3 is very well defined and needs no other representation. Besides, the question arises of what does 0.333... mean. If it means the limit, then that is quite absurd, because the limit is well defined, that is, 1/3. If the representation is of chief consideration, then the only way to get 0.333... is to imagine a bogus infinite sum. The ellipsis does not mean an infinite sum, nor does it mean all the 3s are there. In fact, the limit 1/3 does not care if the 3s are all there or even there at all! > > In the same light, one can't write 3.14159... and call it pi. This too is meaningless nonsense. Consider that no rational number approximation of pi is ever equal to the measure of that incommensurable magnitude pi, which is represented by measuring a circle's circumference using the diameter as a unit. Pi is not a number. > > These bad ideas and definitions have snowballed, so that we have nonsense such as 0.999... = 1 and a bogus mainstream calculus. Rather than correct and revise mathematics, the orangutans who sit atop the academic trash heap simply continue on the same path that has resulted in zero progress the last 150 years. > > For the first and only rigorous formulation of calculus in human history, you will have to discard infinity, infinitesimal, limit theory, real numbers and any other illformed definitions. > > To learn much more than you have learned in all your school years, visit my YT Channel. Also learn about the 13 fallacies in mainstream mathematics. > > The following dishonest academics in the BIG STUPID, have constantly libeled and belittled my claims: > > Gilbert Strang, Professor of mathematics  gilstrang@gmail.com (MIT) > > Jack Huizenga, Professor of mathematics  huizenga@psu.edu (HARVARD) > > Anders Kaesorg, PhD student  andersk@mit.edu (MIT) > > David Ullrich, Professor of mathematics  david.ullrich@math.okstate.edu (David Ullrich) > > I am certain they would love to hear from you with any questions you might have and that is why I have included their email addresses. Also, I am certain you will want to know why they are still holding onto these fallacies. > > Some possible handwaving responses you might get from these academics: > > Mythmatician: But Euler meant the limit. > > No, Euler did not mean the limit. Euler did not say that the fraction 1/(1+a) is the limit and it's very clear he did not think of it as the limit, by the statements: > > 1. The remainder vanishes. > > 2. The infinite sum (S) is the fraction (Lim S). > > To be certain, there is no S without adding up the terms, and to those who are silly enough to argue 0.333... is just the unique representation, well, consider that you can't have 0.333... without adding the terms. When is it unique? After one trillion terms? One light year of terms? Infinity?! Afraid not, infinity is a junk concept. > > Mythmatician: There was no formal word for limit in German. > > Nonsense. The German language had enough words back then (*) to describe the idea, even if there was no official or formal word for limit. Besides, the word Grenz (border) was known in Euler's time. Euler was smart enough to use it, if he really meant to say limit. > > (*) The German nation had just descended from living in the trees only a couple of hundred years before Euler wrote his Elements of Algebra, which are rather primitive when compared with the Works of Archimedes written thousands of years before Euler. > > Mythmatician: But every possible marker/point is covered in the interval (0,1). > > Stupid academics imagine that every point is covered in a given interval and so an infinite sum to them is possible. But this is disproved by Archimedes in this eyeopening video less than 2 minutes. Archimedes used the Archimedan property of rational numbers (not real numbers because there aren't any!) and proof by contradiction to show that the area of the parobolic segment is 4/3 the area of the triangle on the same diameter/chord. Archimedes never recognised any other numbers besides the rational numbers. > > Professor W. Mueckenheim made this comment on sci.math: > > Euler's teacher was Johann Bernoulli, the more conceited and less genial of the Bernoulli brothers (of course being "less genial" than Jakob B does not mean a reproach). Euler was even more genial than both and many others. Nevertheless here he applied the wrong concept. > > John Gabriel is completely correct when he says: > > 1. S = Lim S, is wrong > > 2. The series is not the limit. > > 3. 1/3 cannot be expressed in base 10 because 3 is not a prime factor of 10. > > Unfortunately the contrary belief has lead to the mess of transfinite set theory. > > And this comment on the irrationality of pi: > > It is strange that this clear and concise argument is always circumvented and only irrelevant details are discussed. > > (1) The union of the sequence of intervals [0, (n1)/n) is [0, 1). > > The limit of the sequence of intervals [0, (n1)/n] is [0, 1]. > > (2) The union of the sequence of finite initial segments of the sequence of natural > > numbers {1}, {1, 2}, {1, 2, 3}, ... has less then aleph_0 elements. > > The limit of this sequence (if existing) has aleph_0 elements. > > (3) The sequence of partial sums 3.1, 3.14, 3.141, ... = 3.141... is not irrational. > > The limit of this sequence is irrational. > > It is strange that is always circumvented. Really? Is it? > > Let's summarise a few reasons why it is a bad idea to define S = Lim S: > > [i] It leads to nonequations such as 1/3 = 0.333... > > In algebra, we use the equality sign between numbers. One might say that 0.333... is the number 1/3, because it represents the limit of the series 0.3+0.03+0.003+..., but the problem with this approach, is that academics misguidedly try to perform "infinite" arithmetic using 0.333... and arrive at further absurd results such as 1 = 0.999... To say 0.333... is the limit, is like decreeing that 1/3 = 0.333... In mathematics there is no place for rules or decrees, only logic and common sense. > > [ii] Many academics get the wrong idea that 0.333... is actually an infinite sum, which is obviously impossible. Then their colleagues will deride them by claiming that it's not an infinite sum, only a representation of 1/3. Well, one cannot arrive at this representation without the fallacy of infinite sum. The representation is not a result of long division because long division is a finite process. There is also confusion among academics about the Euclidean algorithm and the long division algorithm. These are not the same! > > [iii] A number theorem in mathematics states that given any p/q and base b, it is not possible to represent p/q in that base b, unless b contains all the prime factors of q. In order to claim that 1/3 can be represented in base 10, you need to find an m and n, such that 1/3 = m / (10^n), where m and n are both integers. Good luck! One would think that on this theorem alone, academics would have been smart enough to realise that S = Lim S is a very bad idea indeed. > > [iv] This leads to numerous other wrong ideas with respect to set theory and is a major time waster with no practical application in science, technology or engineering. The only numbers ever used by humans are the rational numbers. There is no such thing as a "real" number. It is an illusion and a myth. > > Confusion in the mainstream > > But why has the mainstream never realised this blunder you might ask. Well, before you ask this, you may be surprised to know that mainstream academics do not even understand the advocates of this wrong theory, to wit, Rudin on page 59 of his third edition repeats exactly what Euler stated, that is, S = Lim S. > > Rudin's analysis textbook has been the De Facto real analysis textbook used in most courses on real analysis. There is also an irony in the name "real analysis" because there is no valid construction of real number.
An infinite sum is a limit of finite sums. 0.333... is an infinite sum, and an infinite sum is a limit. This is how infinite sums are defined.



