The Rank Holders in Mathematics, Professors in Universities all agree with above Proof as in the reference book.

Since epsilon is arbitrary means epsilon can be very small. But it never guarantee epsilon is zero. In assumption it is clearly specified epsilon is greater than zero.

So the equality cannot be used. Approximately equal is the more perfect argument.

The logic explained here is very clear and convincing to a graduate student in Mathematics. But TRANCE effect in Mathematics Classes and fear to question the Professors rules the world. This is an error and can be corrected by using " approximately equal to" instead equality of L' and L".

But Uniqueness then fails.

The modified additional Steps Required in the Proof.

The modulus value of difference between L' and L" is less than any positive real number. That means Modulus value of difference between L' and L" is an infinitesimal number. Zero is considered as the only real and infinitesimal number. Hence

I L'-L"I = 0.

Therefore L' = L".

I expect Royalty for this additional Steps.

Nishad T M, MSc, MPhil, ( PhD)

Research Investigation Officer

Council for Technology & Science, Kerala

Assistant Professor( Under APJ KTU)& Research Scholar( Under BDU)

Sir, Mr George, I know well what you explained. But mathematical steps are required. We cant write in Proof, "as epsilon approach zero" because then also epsilon is not zero. There is a small positive number that represents the difference of L' and L". So L' will never equal to L". To get a clear idea, or to convey this to students the steps I explained must be included. The difference is infinitesimal, Zero is only real and infinite simal So the difference must be zero.

I L' - L"I =0

implies next step L' = L". This step is required for accuracy and to say unique.

From the existing proof in the reference books, we shall conclude only L' is approximately equal to L".

Please Share this with mathematicians in your circle.

Thanks

The epsilon-delta arguments have a long history in mathematics based on the underlying concept of completeness. When an argument starts of "let epsilon >0 " what are we actually doing. Lets take an example namely the definition of the derivative. For a function of the appropriate conditions, f'(x)=limit as epsilon->0 of (f(x+epsilon)-f(x))/epsilon. Why when epsilon >0, because if epsilon equaled = 0 then the statement is undefined and makes on sense. However, no one questions what f'(x) is. It is the limit. In your example a similar argument shows that the two solutions are actually arbitrary close so in the limit since epsilon is arbitrary - they are equal. The use of this type of argument goes back to Cauchy who introduced it in 1823. The use of these types of arguments evolved over time. Since "infinitesimals" are part of the real number system dy/dx can not really be defined, the approach of "let epsilon >0" evolved to treat limits.

https://arxiv.org/pdf/1502.06942.pdf

The proof in Bartle and Sherbert is fine.

The proof is fine.

It is based on the following fact:

If x < epsilon for every positive epsilon, then x ≤ 0.

This is easily proved by contradiction.

Suppose that x < epsilon for every positive epsilon and that x is not less or equal than 0.

Then x>0, hence for epsilon=x/2 it is epsilon>0 and epsilon

Respected TRUMAN PREVATT,-Since "infinitesimals" are part of the real number system dy/dx can not really be defined.

Limit x tends to 0, Here Limit brings clarity to dy/dx. In proofs of Real Analysis Theorems the arguments "Since epsilon is arbitrary, As epsilon tends to zero, etc" is not bounded by a condition Limit. So there we need more accuracy.

Infinite simals are not real numbers. Only 0 is infinitesimal and real number.

This issue can be solved by bringing modified proof to the statement-The smallest positive real number doest not exist. See the link to know the proof

.https://www.ijmttjournal.org/archive/ijmtt-v67i8p508

Respected GIOVANNI DORE, Please go through above link. Your argument is similar to the existing proof for the statement Smallest Positive Real number does not exist. If that is modified as per the explanation in the above article,(Please go through the link), and the additional steps is added ( as I mentioned in this Question )then this mistake can be cleared.

I hope all Mathematicians and mathematics students will realize this truth and accept the modifications.

I am actually pointing the mistake that has been following in maths world for centuries.

To support my Modifications, I wish to bring all of your attention to the following articles.

https://www.ijmttjournal.org/archive/ijmtt-v67i9p502

https://www.ijmttjournal.org/archive/ijmtt-v67i7p521

If you feel convinced, why do you hesitate to accept the modifications? If my modifications is accepted, I can expect Royalty to include the change that I suggested in Real Analysis.

If I get the Royalties and Rewards now, it will be useful to me. If it comes after long time in my older age or after death, it will not be useful to me. Why all hesitate to accept the modifications required?

Actually Professeur Giovanni Dore's argument can be generalized, with a similar proof : for any two numbers a and b, we have $a \leq b$ iff for every positive number $r$, $a < b + r$.

Respected Lahbib,

The error whatbI pointed can be seen in the Article

https://www.researchgate.net/deref/https%3A%2F%2Fwww.ijmttjournal.org%2Farchive%2Fijmtt-v67i9p502

In my previous comments I had a critical typo, "Since "infinitesimals" are part of the real number system " should have read "Since "infinitesimals" are NOT part of the real number system.

Historically calculus was fraught with philosophical debates about the meaning and logical validity of "fuxions" and "infinitesimals." The solution was developed by Cauchy and the "delta-eplison" arguments and the completeness of the real number system. However, Leibniz imagined a number field containing the reals which also contained the "infinitesimals." In the 1950's and 1960's Abraham Robinson showed such a number field existed, the hyperreal, and he developed the field of Nonstandard Analysis based on the hyperreal number field. https://en.wikipedia.org/wiki/Nonstandard_analysis

While the nonstandard analysis eliminates the philosophical objection to the uniqueness theorem of the original poster - it has not resulted in any new results that could not be ascertained by the "epsilon-delta" approach.

https://mathoverflow.net/questions/16312/how-helpful-is-non-standard-analysis

Respected Truman Prevatt,

***While the nonstandard analysis eliminates the philosophical objection to the uniqueness theorem of the original poster****

Please explain How it eliminates?

The poster's objection was addressed earlier by simply applying to a= norm(f1-f2) a simple argument. Assume that a>0 and chose epsilon= a/2. Then there is a contradiction since a is no longer less than epsilon hence a=0 hence f1=f2, that is uniqueness. QED. That argument should not need to be added especially in a text that this level as it is implicit, no more than one needs to prove the rationals are dense in the reals or that the real numbers are complete every time the fact is used.

Nonstandard Analysis is equivalent to standard analysis. It adds no new results nor any contradictions when compared to not expanding the real numbers to the hyperreal numbers. It simply gives an intuitive meaning to dx, dy, etc., and might give economy in proofs. While there have been criticisms of expanding the reals to the hyperreals and non standard analysis in general - in reality such approaches are more at home in mathematical logic than analysis since they simply replace the epsilon-delta approach of Cauchy with something basically equivalent.

https://en.wikipedia.org/wiki/Criticism_of_nonstandard_analysis

One of the basic criticisms is while it gives meaning to dx, dy and dx/dy, nonstandard analysis does not answer any new questions with less economy.

Repected Truman

Then there is a contradiction since a is no longer less than epsilon hence a=0 hence f1=f2

Here you committed a mistake.

Suppose 'a' where the smallest epsilon then a/2 will not be a real number. Suppose You are saying about Length of Pens. Assume Pen P has the available small length. Then showing a Pencil whose length is smaller than the Pen, how can you bring a contradiction?

If epsilon is the smallest, then half the epsilon in magnitude it is less than epsilon but in its nature, it cant be a real number.

Expecting your reply.

Dear Nishad T M , the argument you believe is not correct is ubiquitous in the book of Bartle & Sherbert; it appears in many places, not only in the page you mention. It is a strong argument that, as Giovanni Dore noted, it works by contradiction and, as Truman Prevatt noted, it hinges upon the completeness – or intuitively speaking, the continuum - of the real numbers.

Given any real number, there does not exist another real number that is closest/nearest to that number; otherwise there would be “gaps” between the reals.

The answer of Giovanni Dore explains the argument.

However, I would like to add that for the argument to hold, the examined numbers should be fixed; i.e. they should not depend on epsilon; otherwise the argument does not hold. For instance, the difference 1/n – 0 can become less than any epsilon greater than zero and yet it can be positive, because, in this case, 1/n changes with epsilon; this is how the concept of limit works. But if the given numbers do not depend on the choice of epsilon, as it happens in the example you mention in your question, then they must be equal.

Dear Spiros,

If a number k is less than every positive real number epsilon , then what about nature of k? Is it real or infinitesimal?

What is definition of infinitesimal.

In the added file, Left hand side of in equality is an infinitesimal and Right hand side is positive real number.

Can we use equality and inequality between two numbers in which one is infinite simal and the other is real number?

Dear Nishad, infinitesimals do not exist in the framework set by the book you are referring to. You do not need to introduce infinitesimals to understand that the argument you believe is wrong is actually true and valid.

You need to understand what proof by contradiction means and what the completeness of reals means. If these two have been understood, then the explanation given by Giovanni Dore is clear; which is also the explanation given in the book itself, when the said argument is introduced, in the first chapters of the book.

Sir,

Method of Contradiction is not better Proof while there are more possibilities. It works only if there is only True or False type possibility.

To Prove square root of 2 is irrational, its better. Because either rational or irrational.

But when the definition of infinite simal numbers already exist in mathematics, we cant apply method of contradiction on numbers so close to zero.I explained that in a simple article. Just few minutes is enough to grasp it.

The article is available free of cost in following link.

https://www.ijmttjournal.org/archive/ijmtt-v67i8p508

Few minutes if you spend, I hope you may get an unexpected but valuable insight.

Nishad T M , every mathematical statement can be either false or true; one of these two, not both; there does not exist a third option.

The argument you believe is false can be either false or true; one of these two, not both. So, proof by contradiction is applicable.

In the 1800's leading up to 1900, the logical foundations of mathematics were not well established and not well understood. Many results were later found wrong - for example the results from the Italian school of algebraic geometry - while their intuition may have seem good, their logical foundations of their proofs were not.

https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry

W. L. Chow often talked about how one's intuition is only a guide but if you believe it too much it will lead you astray. in 1913 Bertrand Russel and Alfred North Whitehead publish the first of the three volume series, Principia Mathematica which established the logical foundations on mathematics. By establishing these foundations, mathematics was able to flourish and it put an end to the era of intricate constructive proofs which lacked logical rigor that lead to the mistakes in the Italian school of algebraic geometry. Hilbert's Nullstellensatz, Hilbert Basis Theorem, was proved to address Gordon's problem the first foundational theorem of the modern days of mathematics that established the relationships between algebra and geometry. After the Nullstellensatz, It was totally non-constructive and depended on the law of the excluded middle - establish a foundation logical axiom by Russel and Whitehead. That is for every proposition then either it is true or it's negation is true.

At the time of Hilbert's solution of Gordan's problem, Gordan himself stated "Das ist nicht Mathematik. Das ist Theologie." (This is no Mathematics. This is Theology). Of course even Gordan had to admit he was wrong later as Hilbert released another paper that established an estimate on the bound of the size of the basis. The Nullstellensatz was instrumental in revolutionizing the way mathematics was done and the Principia Mathematica revolutionized the logical foundations of mathematics.

The law of the excluded middle is the foundation of proof by contradiction. It seems the OP should spend some time reviewing the foundational aspects of mathematical logic.

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