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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Mon, 2 Sep 2024 22:37:02 +0200
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On 02.09.2024 19:19, Richard Damon wrote:
> On 9/2/24 1:07 PM, WM wrote:
>> How many different unit fractions are lessorequal than all unit 
>> fractions? The correct answer is: one unit fraction. If you claim more 
>> than one (two or three or infintely many), then these more must be 
>> equal. But different unit fractions are different and not equal to 
>> each other.
>>
>> Another answer is that no unit fraction is lessorequal than all unit 
>> fractions. That means the function NUF(x)
>> Number of UnitFractions between 0 and x > 0
>> with NUF(0) = 0 will never increase but stay at 0. There are no unit 
>> fractions existing at all.
>>
>> Therefore there is only the one correct answer given above.
>>
> Nope, because there does not exist AHY unit fraction that is less than 
> or equal to ALL Unit fractions,

Impossible because then NUF will never increase. Then there are no unit 
fractions.

> as any unit fraction you might claim to 
> be that one has a unit fraction smaller than itself, so it wasn't the 
> smallest.

Your argument stems from visible unit fractions but becomes invalid in 
the dark domain.
> 
> The problem with your NUF, is that it is trying to count something from 
> and uncountable end, one that doesn't actually have an end.

The unit fractions end before zero.

Regards, WM