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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: Thu, 5 Sep 2024 22:44:03 +0200
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On 05.09.2024 20:56, Jim Burns wrote:
> On 9/5/2024 9:53 AM, WM wrote:

> Insisting that ω-1 exists and that,
> for b ≠ 0 and β < ω, β-1 exists
> is
> insisting that ω is finite.

No.
> 
> The most frugal explanation of your claim is that
> you simply do not know what 'finite' means.

Finite means that you can count from one end to the other. Infinite 
means that it is impossible to count from one end to the other.

>> Do you believe that it needs a shift to state:
>> All different unit fractions are different.
>> ∀n ∈ ℕ: 1/n - 1/(n+1) > 0
>> I can see no shift.
> 
> It needs a shift to conclude from
> ( for each ⅟j: there is ⅟k≠⅟j: ⅟k < ⅟j
> that
> ( there is ⅟k: for each ⅟j≠⅟k: ⅟k < ⅟j
> 
> Have you evolved on that topic?

You are mistaken. I do not conclude the latter from the former. I 
conclude the latter from the fact that NUF(0) = 0 and NUF(x>0) > 0 and 
never, at no x, NUF can increase by more than 1.

Try to understand that.

Regards, WM