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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Does the number of nines increase?
Date: Tue, 16 Jul 2024 05:08:33 +0200
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Am 16.07.2024 um 04:55 schrieb Moebius:
> Repost:
> 
> Am 16.07.2024 um 04:02 schrieb Chris M. Thomasson:
>> On 7/15/2024 6:57 PM, Moebius wrote:
> 
>>> See? 😛
> 
>> I see that you increased the granularity from natural numbers into the unit fractions...

> Yeah, the real numbers comprise the naturals, the integers, the unit 
> fractons, the rational numbers, ..., you know. 🙂
> 
>> Wrt enumeration unit fractions I like to go from 1/1, to 1/2, to 1/3, ect...
> 
> 
> You may like to do that, still:
> 
> 0 < ... < 1/3 < 1/2 < 1/1.
> 
> Meaning: Concening the < relation as _defined on the reals_ (as well on 
> the rationals) 1/3 is SMALLER than 1/2 and 1/2 is smaller than 1/1. In 
> general: 1/(n+1) is smaller than 1/n.
> 
>> Is that wrong?
> 
> Nope. You may define a SEQUENCE (of unit fractions):
> 
> (1/1, 1/2, 1/3, ...)
> 
> Here (referring to these sequence) 1/1 is "before", say, 1/2. 🙂
> 
>> we have to think of a a smallest unit fraction, WM world, right?
> 
> Right. There simply is no such unit fraction because for each and every 
> unit fraction u: 1/(1/u + 1) is a unit fraction that is smaller than u.

Of course, we may consider the set {1/1, 1/2, 1/3, ...} of all unit 
fractions and define a certain order << on it, such that

1/1 << 1/2 << 1/3 << ... .

But this is NOT the order WM is referring to. WM is referring to the 
usual order < as defined on the reals (or rationals). There

0 < ... < 1/3 < 1/2 < 1/1.

Nuff said. (Ufff...)