Me
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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


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*.

