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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Tue, 10 Sep 2024 13:33:49 -0700
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On 9/10/2024 12:30 PM, Moebius wrote:
> Am 10.09.2024 um 20:38 schrieb Chris M. Thomasson:
>> On 9/10/2024 11:35 AM, Chris M. Thomasson wrote:
>>> On 9/9/2024 5:32 PM, Moebius wrote:
>>>> Am 10.09.2024 um 01:08 schrieb Chris M. Thomasson:
>>>>
>>>>> Take the gap between:
>>>>>
>>>>> 1/1 and 1/2. There are infinitely many unit fractions that are 
>>>>> small enough to fit within that gap.
>>>>
>>>> Oh, really?! Could you name (just) o n e? :-)
>>>
>>> 1/4?
>>
>> 1/4 is between 1/1 and 1/2:
>>
>> 1/1---->1/4---->1/2
> 
> lol. If you say so. :-)
> 
>> So, it's not _strict_ 
> 
> Actually, it's not between 1/2 and 1/1 at all (if we presuppose the 
> usual order < on the rational numbers).

The gap from 1/1 to 1/2 is 1/2 so there are infinity many unit fractions 
that are smaller than 1/2.

So, say

1/1----->(1/4 + 1/4 + 1/8 + 1/8)----->1/2

Wrt the number line. However, the gap is only 1/2 and 1/4 is smaller 
than that... There are infinite unit fractions smaller than 1/2, right?

Going off the rails with this line of thought? Heck I can even make a 
fractal on it based on infinite unit fractions in between any gap.