Re: Replacement of Cardinality

Am Fri, 16 Aug 2024 16:45:29 +0000 schrieb WM:

Le 15/08/2024 à 19:01, Moebius a écrit :
 

Assume that there is an x e IR such that NUF(x) = 1. Let x0 e IR such
that NUF(x0) = 1. This means that there is exactly one unit fraction u
such that u < x0. Let's call this unit fraction u0. Then (by
definition)
there is a (actually exactly one) natural number n such that u0 = 1/n.
Let n0 e IN such that u0 = 1/n0. But then (again by definition) 1/(n0 +
1) is an unit fraction which is smaller than u0 and hence smaller than
x0. Hence NUF(x0) > 1. Contradiction!

We can reduce the interval (0, x) c [0, 1] such that x converges to 0.
Then the number of unit fractions diminishes. Finally there is none
remaining. But never, for no interval (0, x), more than one unit
fraction is lost. Therefore there is only one last unit fraction.

It does not diminish, there are always infinitely many.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.