Cont: Deeper than primes - Continuation 2

[doronshadmi]

All you have done so far is repeatedly claimed the "all in one step" summation is possible. You have yet to show how. Show us how it is possible using the sequence (1, -1/2, 1/3, -1/4, 1/5, ...).

Since all |N| terms of sequence (1, -1/2, 1/3, -1/4, 1/5, ...) are included in it by a common principle, we are using these facts by add them in one step.

This reasoning is equivalent to the mathematical ability to define the value of |N| as the number of elements in {1, 2, 3, ...}, without counting them step by step (which means: only one step is needed in order to define the value of |N|).

 

[jsfisher]

So please show such an element.

How convenient. You skipped over the part of my post asking you to justify your statement and instead attempted to obligate me for something not relevant to that.

For your reference, here is that important part again:

You are the one that made the bogus claim that "0 < |a_n - L|" was some part of the definition of 'limit'. It is for you to either support that bogus claim or retract it.

 

[doronshadmi]

regarding the definition of 'limit'.

There is no definition of limit unless by this definition |a_n - L| = 0.

So by using a_n, which is definitely one of the terms of the infinite sequence of partial sums (s₁, s₂, s₃, ...), please demonstrate the case of |a_n - L| = 0.

 

[jsfisher]

Since all |N| terms of sequence (1, -1/2, 1/3, -1/4, 1/5, ...) are included in it by a common principle, we are using these facts by add them in one step.

This reasoning is equivalent to the mathematical ability to define value of |N| as the number of elements in {1, 2, 3, ...}, without counting them step by step (which means: only one step is needed in order to define the value of |N|).

You continue to assert that the summation can be done in one step, but you never show how it can be done. If you cannot show how such a summation can be taken, that you can make no claim about what the sum is.

You then have no basis to claim 0.999... is different from 1.

 

[doronshadmi]

How convenient. ...

So you can't show such an element, where without such an element your definition does not hold water.

 

[doronshadmi]

You continue to assert ...

You continue to ignore the simple facts about the mathematical ability to do that, exactly as demonstrated in the case of |N|.

Moreover, you can't demonstrate how |a_n - L| = 0, and therefore your definition of limit does not hold water.

 

[jsfisher]

There is no definition of limit unless by this definition |a_n - L| = 0.

Prove it. And proving it would require you work from the actual definition for 'limit', not what you imagine in might be like. While your there, please highlight were in that definition "0 < |a_n - L|" is to be found.

You claimed "0 < |a_n - L|" was some part of the definition of 'limit'. It is for you to either support that bogus claim or retract it.

 

[doronshadmi]

Prove it. And proving it would require you work from the actual definition for 'limit', not what you imagine in might be like.

Exactly.

So by using a_n, which is definitely one of the terms of the infinite sequence of partial sums (s₁, s₂, s₃, ...), please demonstrate the case of |a_n - L| = 0.

Without this demonstration you have no actual definition for limit.

So please support your which I can in http://www.internationalskeptics.com/forums/showpost.php?p=10644992&postcount=479.

 

[jsfisher]

Exactly.

So by using a_n, which is definitely one of the terms of the infinite sequence of partial sums (s₁, s₂, s₃, ...), please demonstrate...

You continue to try to shift the burden. Your claim; your burden. You claimed "0 < |a_n - L|" was a part of the definition of 'limit'. Put up or shut up.

 

[doronshadmi]

You continue to try to shift the burden. Your claim; your burden. You claimed "0 < |a_n - L|" was a part of the definition of 'limit'. Put up or shut up.

You are still missing it.

Without clearly demonstrate that by your definition |a_n - L| = 0 you are left only with 0 < |a_n - L|, and in this case your definition of limit is no more than your fantasy, that is based on forcing the finite on the infinite.

So put up your which I can (exactly as explained and requested in http://www.internationalskeptics.com/forums/showpost.php?p=10645112&postcount=488) or shut up.

 

[jsfisher]

You continue to try to shift the burden. Your claim; your burden. You claimed "0 < |a_n - L|" was a part of the definition of 'limit'. Put up or shut up.

You are still missing it.

You claimed "0 < |a_n - L|" was a part of the definition of 'limit'. Either it is, or it isn't. (It isn't.) It does not matter at all what I can demonstrate about the definition; it is only your claim that is at issue. Either your claim is true, or it is not. (It is not.)

Moreover, given the effort you are expending to evade your responsibility, one might reasonably conclude that it was not an accidental misstatement. You have knowingly made false statements on previous occasions. Is this yet another one?

What will it be, doronshadmi: Will you retract the claim or continue to let the lie stand?

 

[doronshadmi]

You claimed "0 < |a_n - L|" was a part of the definition of 'limit'. Either it is, or it isn't. (It isn't.)

You are wrong, 0 < |a_n - L| is an inseparable part of the standard definition of limit, exactly as clearly given in http://www.internationalskeptics.com/forums/showpost.php?p=10647618&postcount=490 including the content of its link.

As a result the standard definition of limit does not hold water.

 

[jsfisher]

You are wrong, 0 < |a_n - L| is an inseparable part of the standard definition of limit, exactly as clearly given in http://www.internationalskeptics.com/forums/showpost.php?p=10647618&postcount=490.

Your linked-to post does not include the definition of 'limit'. Moreover, you need to present the definition of 'limit' from some actual authority. Your own posts do not qualify. Wikipedia, on the other hand, would be acceptable.

This should be simply if what you said had been the truth. You would need only quote the meaning of 'limit' (actually, only the portion of its definition applicable to infinite series would be required) and then highlight the text that matches your claim.

Unfortunately for you, what you said was not the truth. Your continued defense of it only deepens your lie and discredits you position. Are you not embarrassed by your failure?

 

[doronshadmi]

Your linked-to post does not include the definition of 'limit'. Moreover, you need to present the definition of 'limit' from some actual authority. Your own posts do not qualify. Wikipedia, on the other hand, would be acceptable.

This should be simply if what you said had been the truth. You would need only quote the meaning of 'limit' (actually, only the portion of its definition applicable to infinite series would be required) and then highlight the text that matches your claim.

Unfortunately for you, what you said was not the truth. Your continued defense of it only deepens your lie and discredits you position. Are you not embarrassed by your failure?

The standard definition of limit is: for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |a_n - L| < ε.

So a_n is some term in the infinite sequence of partial sums (s₁, s₂, s₃, ...) that it is arbitrary closer to L such that |a_n - L| < ε, where ε > 0.

Since a_n is some partial sum that at its best only tends to L (for example: L=1 and a_n is some term of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) that it is arbitrary closer to 1) it is obviously clear that |a_n - L| > 0. Since you claim that |a_n - L| = 0 is also possible by the standard definition of limit, then please provide such case, or shut up (about "shut up" please look at http://www.internationalskeptics.com/forums/showpost.php?p=10647618&postcount=490).

Moreover, no one of the terms of the infinite sequence of partial sums (s₁, s₂, s₃, ...) (for example: (0.9₁₀, 0.99₁₀, 0.999₁₀, ...)), where each one of them has a finite amount of detailed symbols, can actually be used in order to conclude anything about a number that its sum is defined by infinite amount of detailed symbols (for example: number 0.999...₁₀, where each one the terms of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) is no more than its partial finite sum).

Mathematics gives meaning to infinite series by defining it to be the limit of the sequence of partial sums.

In other words, jsfisher, the standard definition of limit does not hold water.

Are you not embarrassed by your failure to realize that?

 

[doronshadmi]

So, for example, 0.999...₁₀ is not identical to 1.000... by the standard definition of limit, because in both cases (s₁, s₂, s₃, ...) (a sequence of partial sums) is useless, meaningless, etc., in order to define 0.999...₁₀ as identical to 1.000...

 

[jsfisher]

The standard definition of limit is: for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |a_n - L| < ε.

Not quite. You left out a lot of important stuff, but imprecision is your specialty, so let's just correct what you have and work from there:

A real-valued sequence (a₁, a₂, a₃, ...) has a limit, L, if and only if for all real numbers ε > 0 there exists a natural number N such that for all n > N, |a_n - L| < ε.

Please notice that the choice for variable names was arbitrary and not fixed by the definition. References to a_n or L in isolation have no special affinity to or automatic association with the limit definition.

Now to the point: No where in the definition is there any requirement for 0 < |a_n - L|. Anyone who insisted otherwise would have been lying, ignorant of the actual definition, or both. Doronshadmi, which were you?

 

[doronshadmi]

A real-valued sequence (a₁, a₂, a₃, ...) has a limit, L, if and only if for all real numbers ε > 0 there exists a natural number N such that for all n > N, |a_n - L| < ε.

Please notice that the choice for variable names was arbitrary and not fixed by the definition. References to a_n or L in isolation have no special affinity to or automatic association with the limit definition.

Please give me a break, I know that the names of the variables are irrelevant to a given definition, so you still have no point.

Moreover, your version still uses partial sums, and this is exactly my argument against the standard definition of limit.

Now to the point: No where in the definition is there any requirement for 0 < |a_n - L|.

Dear jsfisher, 0 < |a_n - L| is inevitable by the standard definition exactly because a_n is used for terms of an infinite sequence of |N| partial sums that no one of them is identical to L.

As long as you do not show (by using your standard definition, which is entirely based on partial sums) that also 0 = |a_n - L|, you simply have no actual definition of limit, which is actual only if 0 = |a_n - L|.

So by using your sequence of partial sums, please demonstrate the 0 = |a_n - L|, because without it you have no more than some arbitrary agreement among a group of people that is not based on any reasoning.

 

[jsfisher]

Taking this as the backdrop:

A real-valued sequence (a₁, a₂, a₃, ...) has a limit, L, if and only if for all real numbers ε > 0 there exists a natural number N such that for all n > N, |a_n - L| < ε.

...let's continue.

So a_n is some term in the infinite sequence of partial sums (s₁, s₂, s₃, ...)

You would appear less confused if you were consistent in you notation. The elements of your sequence of partial sums are all labeled of the form s_i. Your "some term" should be s_n.

...that it is arbitrary closer to L such that |a_n - L| < ε, where ε > 0.

What's L? 'Arbitary closer [sic]' is a comparative. Arbitrarily closer than what?

To the casual observer, doronshadmi, it might appear that you are somehow trying to turn the definition of limit inside-out through some act of torture. It might appear that you believe the definition somehow designates some distinguished element, s_n, that has some important properties.

It doesn't, and of course you wouldn't....would you?

Since a_n is some partial sum that at its best only tends to L

Well, that answers that. a_n s_n doesn't tend toward anything.

...(for example: L=1 and a_n is some term of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) that it is arbitrary closer to 1)

For the sequence (0.9, 0.99, 0.999, ...) there is no element that is arbitrary closer [sic] to 1.

...it is obviously clear that |a_n - L| > 0.

Are you trying to say that for all elements of the sequence (0.9, 0.99, 0.999, ...) there is a positive difference between the element and the limit of the sequence? That was never in dispute. Why do you bring it up? It is not relevant to either the limit of the sequence nor the definition of 'limit'.

Since you claim that |a_n - L| = 0 is also possible by the standard definition of limit

It is not a question of possibility by the definition of limit. The definition is silent on this matter.

...then please provide such case

Are you really so narrowly focused, obsessed actually, on 0.999... that you fail to see the obvious?

One example: (1, 0, 1/2, 0, 1/3, 0, 1/4, 0, 1/5, ...)

...
In other words, jsfisher, the standard definition of limit does not hold water.

Are you not embarrassed by your failure to realize that?

The facts would lead one to a different conclusion.

 

[jsfisher]

Please give me a break, I know that the names of the variables are irrelevant to a given definition, so you still have no point.

You have repeated posted "0 < |a_n - L|" and expected an instant and automatic association with the definition of 'limit'. So, yes, there was a point.

Moreover, your version still uses partial sums, and this is exactly my argument against the standard definition of limit.

Really? Exactly where does my version use partial sums? For your convenience, here's the definition again. Please just highlight were I make any mention of partial sums whatsoever.

A real-valued sequence (a₁, a₂, a₃, ...) has a limit, L, if and only if for all real numbers ε > 0 there exists a natural number N such that for all n > N, |a_n - L| < ε.​

 

[doronshadmi]

Are you trying to say that for all elements of the sequence (0.9, 0.99, 0.999, ...) there is a positive difference between the element and the limit of the sequence? That was never in dispute. Why do you bring it up? It is not relevant to either the limit of the sequence nor the definition of 'limit'.

It is most relevant to the limit of a given infinite sequence and for a valid definition of limit.

The definition is silent on this matter.

Absolutely not, by your definition 0 < |a_n - L| is inevitable.

Are you really so narrowly focused, obsessed actually, on 0.999... that you fail to see the obvious?

One example: (1, 0, 1/2, 0, 1/3, 0, 1/4, 0, 1/5, ...)

Since it diverges it is obviously 0 < |a_n - L|, so you still have no case.