Re: Replacement of Cardinality

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Sujet : Re: Replacement of Cardinality
De : wolfgang.mueckenheim (at) *nospam* tha.de (WM)
Groupes : sci.logic sci.math
Date : 16. Aug 2024, 17:45:29
Autres entêtes
Organisation : Nemoweb
Message-ID : <UMzq2D4JrBFmHiWT8a6U533RZeg@jntp>
References : 1 2 3 4 5 6 7 8 9 10
User-Agent : Nemo/0.999a
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.
 Regards, WM

Date Sujet#  Auteur
15 Aug 24 * Re: Replacement of Cardinality47Moebius
15 Aug 24 +* Re: Replacement of Cardinality9Python
15 Aug 24 i+- Re: Replacement of Cardinality1Moebius
15 Aug 24 i+* Re: Replacement of Cardinality2Chris M. Thomasson
16 Aug 24 ii`- Re: Replacement of Cardinality1Moebius
16 Aug 24 i`* Re: Replacement of Cardinality5WM
16 Aug 24 i +- Re: Replacement of Cardinality1Richard Damon
16 Aug 24 i `* Re: Replacement of Cardinality3Chris M. Thomasson
16 Aug 24 i  `* Re: Replacement of Cardinality2FromTheRafters
17 Aug 24 i   `- Re: Replacement of Cardinality1Chris M. Thomasson
15 Aug 24 +* Re: Replacement of Cardinality2Moebius
15 Aug 24 i`- Re: Replacement of Cardinality1Moebius
16 Aug 24 `* Re: Replacement of Cardinality35WM
16 Aug 24  `* Re: Replacement of Cardinality34joes
16 Aug 24   `* Re: Replacement of Cardinality33WM
16 Aug 24    +* Re: Replacement of Cardinality2Richard Damon
16 Aug 24    i`- Re: Replacement of Cardinality1Chris M. Thomasson
16 Aug 24    +* Re: Replacement of Cardinality12Jim Burns
16 Aug 24    i+* Re: Replacement of Cardinality3Chris M. Thomasson
16 Aug 24    ii`* Re: Replacement of Cardinality2Jim Burns
17 Aug 24    ii `- Re: Replacement of Cardinality1Chris M. Thomasson
17 Aug 24    i`* Re: Replacement of Cardinality8WM
17 Aug 24    i `* Re: Replacement of Cardinality7Jim Burns
18 Aug 24    i  +* Re: Replacement of Cardinality3Moebius
19 Aug 24    i  i`* Re: Replacement of Cardinality2WM
19 Aug 24    i  i `- Re: Replacement of Cardinality1Chris M. Thomasson
19 Aug 24    i  `* Re: Replacement of Cardinality3WM
19 Aug 24    i   `* Re: Replacement of Cardinality2Jim Burns
20 Aug 24    i    `- Re: Replacement of Cardinality1WM
16 Aug 24    `* Re: Replacement of Cardinality18joes
16 Aug 24     +- Re: Replacement of Cardinality1Moebius
16 Aug 24     +* Re: Replacement of Cardinality2Richard Damon
17 Aug 24     i`- Re: Replacement of Cardinality1Chris M. Thomasson
16 Aug 24     +* Re: Replacement of Cardinality2FromTheRafters
17 Aug 24     i`- Re: Replacement of Cardinality1Chris M. Thomasson
17 Aug 24     `* Re: Replacement of Cardinality12WM
17 Aug 24      +* Re: Replacement of Cardinality6Richard Damon
17 Aug 24      i+- Re: Replacement of Cardinality1Chris M. Thomasson
19 Aug 24      i`* Re: Replacement of Cardinality4WM
19 Aug 24      i `* Re: Replacement of Cardinality3Richard Damon
19 Aug 24      i  `* Re: Replacement of Cardinality2WM
19 Aug 24      i   `- Re: Replacement of Cardinality1Chris M. Thomasson
18 Aug 24      `* Re: Replacement of Cardinality5joes
19 Aug 24       `* Re: Replacement of Cardinality4WM
19 Aug 24        +- Re: Replacement of Cardinality1Jim Burns
19 Aug 24        `* Re: Replacement of Cardinality2Chris M. Thomasson
20 Aug 24         `- Re: Replacement of Cardinality1WM

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