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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?
Date: Mon, 2 Sep 2024 17:43:45 -0400
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On 9/2/24 4:37 PM, WM wrote:
> 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.

Which just shows the error in the "definition" of NUF.

> 
>> 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.

But all the unit fractions are visible. You agreed to that you self as 
you said if n is visible, so will be n+1.

Thus, there is no smallest visible unit fraction as there can't be a 
last one.

>>
>> 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.

No, they don't end, they have a bound that is outside of themselves, but 
have no final member.

> 
> Regards, WM
>