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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

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Me

Posts: 1,340
Registered: 1/23/16
Re: Teaching mainstream morons about their own flawed theories:
Every Cauchy sequence of real numbers converges to a limit.

Posted: Sep 27, 2017 6:49 PM
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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*

Man, these lectures are dealing with real analysis.

> In spite of this, you will get thousands of [reasonable people,
> knowing their stuff] (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, staring just with IN (i.e. the Peano axioms) there's indeed a set Q which (at that point) is NOT "Part of IR, especially since at that point IR has not yet be defined.

See: https://www.kullabs.com/img/note_images/rIk7z5Y1BF9493BD.jpeg

> 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.

> [and] this [just] mean[s] the sequence does not converge

...in the context of the rational numbers. Right!

> ... 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*.




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