Re: There is a first/smallest integer (in Mückenland)

On 7/17/2024 2:11 PM, Chris M. Thomasson wrote:

On 7/17/2024 12:50 PM, Chris M. Thomasson wrote:

On 7/17/2024 10:01 AM, joes wrote:

Am Wed, 17 Jul 2024 15:08:30 +0000 schrieb WM:

Le 17/07/2024 à 16:56, Moebius a écrit :

Am 17.07.2024 um 16:43 schrieb WM:

>

Can you explain how NUF(x) can [jump] from 0 [at x = 0] to [aleph_0]
[at any]
point x [> 0] although all unit fractions are separated by finite
distances [...]

>
Yes, of course: For each and every x e IR, x > 0 there are
countably-infinitely many unit fractions which are <= x. (Hint: No
first one.)

>
Thema verfehlt. The question is: How does NUF(x) increase from 0 to
more? There is a point where NUF is 0 and then it increases. How?

>
>

The same as the sign function. There simply is no such "point", as
there is no least positive number. The distances between unit
fractions get infinitely small.
>

>
Right!

 Wrt sign, there is no least positive number. Take any positive number and multiply it by -1. Therefore there is no least negative number. The negatives and the positives get infinitely small, tending towards zero...

A monkey wrench is so called signed zero. Where positive zero is the least positive number? Fun twists that are besides the point. lol. ;^)

 The abs of the number tending towards zero is interesting. Always a positive distance towards zero... :^)