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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: =?UTF-8?Q?Re=3A_There_is_a_first/smallest_integer_=28in_M=C3=BCcken?=
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Date: Wed, 17 Jul 2024 12:51:21 -0700
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On 7/17/2024 10:17 AM, WM wrote:
> Le 17/07/2024 à 19:01, joes a écrit :
>> Am Wed, 17 Jul 2024 15:08:30 +0000 schrieb WM:
>>> Le 17/07/2024 à 16:56, Moebius a écrit :
>>>> Am 17.07.2024 um 16:43 schrieb WM:
>>>
>>>>> Can you explain how NUF(x) can [jump] from 0 [at x = 0] to [aleph_0]
>>>>> [at any]
>>>>> point x [> 0] although all unit fractions are separated by finite
>>>>> distances [...]
>>>>
>>>> Yes, of course: For each and every x e IR, x > 0 there are
>>>> countably-infinitely many unit fractions which are <= x. (Hint: No
>>>> first one.)
>>>
>>> Thema verfehlt. The question is: How does NUF(x) increase from 0 to
>>> more? There is a point where NUF is 0 and then it increases. How?
>> The same as the sign function.
> 
> No, ℵo finite intervals do not fit between [0, 1] and (0, 1]. The sign 
> function fits.
> 
>> There simply is no such "point", as
>> there is no least positive number. The distances between unit
>> fractions get infinitely small.
> 
> They remain finite in every case.

There are infinitely many of them, and none of them equals zero... 0/1 
is not a unit fraction! Damn it. :^)