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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Fri, 13 Sep 2024 13:12:24 -0400
Organization: i2pn2 (i2pn.org)
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On 9/13/24 12:16 PM, WM wrote:
> On 13.09.2024 08:40, joes wrote:
> 
>> The "end" of an interval is a number such that no element is larger/
>> smaller, defined as the limit, which may or may not be included.
> 
> No, the end is a point, in case of open intervals a dark point. There is 
> no gap on the real axis. All is full of points.
> 
> Regards, WM
>>
> 

So you think, but can't show, because you are just wrong.

Just past the end is a point, (the boundry of the open interval) and 
there is no such thing as two consecutive points in a dense system, so 
there CAN'T be a point at the end of the open interval.

The problem is you don't understand the meaning of there is no "gap" on 
the number line. It doesn't mean that points are next to each other, 
because each has some width to take the space between the two different 
points, but that between any two distinct points is a cloud of more 
points filling the space, all distinct and all unique, but not finitely 
countable.

Thus, we have no points are adjacent, as between them is always more points.