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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? (infinitary)
Date: Thu, 10 Oct 2024 20:18:48 +0200
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On 09.10.2024 19:47, joes wrote:
> Am Wed, 09 Oct 2024 18:56:29 +0200 schrieb WM:
>> Am 09.10.2024 um 18:12 schrieb Alan Mackenzie:
>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>
> Especially since both all segments are infinite,
> and there are infinitely many of them.

Impossible. Infinite endsegments contain almost all numbers. Therefore 
it is not possible that almost all numbers are used as indices.
> 
>>>>>>>> Note: The shrinking endsegments cannot acquire new numbers.
>>>>>>> An end segment is what it is.  It doesn't change.
>>>>>> But the terms of the sequence do. Here is a simple finite example:
>>>>>> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
>>>>>> {2, 3, 4, 5, 6, 7, 8, 9, 10}
>>>>>> {3, 4, 5, 6, 7, 8, 9, 10}
>>>>>> {4, 5, 6, 7, 8, 9, 10}
>>>>>> {5, 6, 7, 8, 9, 10}
>>>>>> {6, 7, 8, 9, 10}
>>>>>> {7, 8, 9, 10}
>>>>>> {8, 9, 10}
>>>>>> {9, 10}
>>>>>> {10}
>>>>>> { } .
>> The completion of the above sets does not change the principle:
>> Non-empty inclusion-monotonic sets like infinite endsegments have a
>> non-empty intersection. All endsegments have an empty intersection.
> You should really be more careful with your phrasing. Intersection with
> what?

The intersection of all endsegments containing at least 3 numbers with 
just these endsegments contains 3 numbers.
The intersection of all endsegments with all endsegments is empty.

Regards, WM