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

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