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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: Replacement of Cardinality
Date: Fri, 16 Aug 2024 16:49:25 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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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.

-- 
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.