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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: Mon, 16 Sep 2024 15:03:45 -0700
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On 9/16/2024 3:02 PM, Ben Bacarisse wrote:
> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
> 
>> On 9/15/2024 9:38 AM, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>
>>>> On 9/14/2024 11:35 AM, joes wrote:
>>>>> Am Sat, 14 Sep 2024 16:01:02 +0200 schrieb WM:
>>>>>> On 14.09.2024 01:05, FromTheRafters wrote:
>>>>>>> WM explained :
>>>>>>
>>>>>>>> No, that is your big mistake. In the interval [0, 1] there is a point
>>>>>>>> next to 0 and a point next to 1, and infinitely many are beteen them.
>>>>>>> Define 'next' in this context.
>>>>>> Two points are next to each other means that no point is between them.
>>>>> Which is the case for no two (different) reals.
>>>>
>>>> Two points on the real line that are different from one another have
>>>> infinite points between them, and so on and so forth. :^)
>>> It might be worth pointing out that any non-trivial interval [a, b] on
>>> the real line (i.e. with b > a) contains an uncountable number of
>>> points.  Constructing the mid point and the quarter points and so on
>>> only shows a countably infinite number of internal points, but giving a
>>> bijection between [a, b] and [0, 1] shows that they have the same
>>> cardinality.
>>
>> Agreed. Afaict, one way to cover all the points is to draw a solid line
>> between two different points p0 and p1 where p0 does not equal p1. We can
>> say the line covers them all? Fair enough? Put two different points on a
>> piece of paper and draw a line from p0 to p1. That line contains infinitely
>> dense points.
> 
> Too much physics for me to comment.
> 

Does it involve zooming into the drawn line with a very powerful 
microscope? ;^)