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

Am 30.10.2024 um 21:04 schrieb Chris M. Thomasson:

On 10/30/2024 12:48 PM, WM wrote:

On 30.10.2024 16:43, FromTheRafters wrote:

on 10/30/2024, WM supposed :

>

Believe what you like without foundation.
If ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 is true, the NUF(x) grows in steps of not more than 1.

>
Wrong.

>
What? ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 ?

 1/1 - 1/2 is greater than zero.
 Wow, what a discovery! lol. ;^/
 1/2 - 1/3 is greater than zero.
1/3 - 1/4 is greater than zero.
... On and on ...
 They never hit zero. Got it?
 No unit [fraction] equals zero.

Yeah, that should be clear from the fact that an unit fraction has "the form" 1/n where n is a natural number (by definition). Since there is no natural number n such that 1/n = 0. :-P (If there were such a natural number, say wm, we would get 1 = 1/wm * wm = 0 * wm = 0.)
Mückenheim's great discovery is that the DISTANCE between two "adjacent" unit fractions is always > 0. Incredible!
For absolutely no reason he now "thinks" that this fact implies a smallest unit fraction. :-)