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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit fractions?
Date: Tue, 10 Sep 2024 15:18:42 -0400
Organization: Peripheral Visions
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Chris M. Thomasson formulated on Tuesday :
> On 9/10/2024 4:09 AM, FromTheRafters wrote:
>> Chris M. Thomasson brought next idea :
>>> On 9/9/2024 11:49 AM, WM wrote:
>>>> On 09.09.2024 17:27, Python wrote:
>>>>> Le 09/09/2024 à 17:15, Prof. Dr. Mückenheim, aka WM a écrit :
>>>>>> On 09.09.2024 16:32, Python wrote:
>>>>>>> Le 09/09/2024 à 12:19, Prof. Dr. Mückenheim, aka WM a écrit :
>>>>
>>>>> 1/n is not in (0, x). Sure. So what? Nevertheless there are Aleph_0 unit
>>>>> fractions in (0, x). No need for 1/n to be there, there far enough other
>>>>> fractions.
>>>>>
>>>> If you cannot understand mathematics consider the simplest logic:
>>>> If a sequence of different real points exists on the positive real axis, 
>>>> then it has a beginning. Otherwise it could be a cloud but not a 
>>>> sequence.
>>>
>>> Oh wow. You need help! Or we do for even giving you the time of day. 
>>> YIKES!!!!!!!!!!
>> 
>> Actually, I think he is close to correct. The thing is, he wants the first 
>> term of the sequence to be last. The reals are not a sequence, but the unit 
>> fractions are and they start with 1/1 not at some imagined other end. You 
>> need a first and likely a next to start.
>
> So, I am thinking there is a unit fraction that fits in the simple gap of 1/1 
> and 1/2:
>
> 1/1----->1/4------>1/2
>
> ?
>
> Fair enough? Is this getting on track or going off the rails?

That's not the same ordered sequence, but as an interval and having a 
measure called distance two of these 1/4 length segments will indeed 
'fit' between the one and the one-half endpoints.

Not the same as inserting an element in an out of order sequence.