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

Le 17/07/2024 à 22:37, joes a écrit :

Am Wed, 17 Jul 2024 17:17:54 +0000 schrieb WM:

Le 17/07/2024 à 19:01, joes a écrit :

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.

No, ℵo finite intervals do not fit between [0, 1] and (0, 1]. The sign
function fits.

Where do you get this requirement from?

ℵo unit fractions occupy ℵo finite intervals. 

Consider the sign function times infinity.

The sign function can change from point 0 to all points of the interval (0, oo) without exception. NUF needs some of these points to acquire ℵo unit fractions.
Regards, WM