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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Sun, 27 Oct 2024 14:00:30 -0400
Organization: i2pn2 (i2pn.org)
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On 10/27/24 12:12 PM, WM wrote:
> Am 27.10.2024 um 14:54 schrieb Moebius:
>> Am 27.10.2024 um 08:38 schrieb WM:
>>
>>> and after NUF(x') = 1 
>>
>> There is no x' e IR such that NUF(x') = 1.
> 
> If all unit fractions are existing, then there is each one. Never two or 
> more are at the same point. Hence there is a single one first.

No two together, but no need for one to be first at the large 'n' end of 
them.

That is the nature of being infinite.

>>
>> Hint: For each and every x e IR, x <= 0: NUF(x) = 0
>> and for each and every x e IR, x > 0: NUF(x) = aleph_0.
> 
> That is blatantly wrong because it would require that ℵo unit fractions 
> exist between 0 and each and every x > 0, i.e., the open interval (0, 1].
> 
> Regards, WM
> 

Right, like is what happens.

It may seem strange to a person stuck in finite logic, but is true when 
you understand how infinity works.