Re: How many different unit fractions are lessorequal than all unit fractions?

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.