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

On 27.10.2024 19:04, joes wrote:

Am Sun, 27 Oct 2024 17:12:23 +0100 schrieb WM:

Am 27.10.2024 um 14:54 schrieb Moebius:

Am 27.10.2024 um 08:38 schrieb WM:
>

and after NUF(x') = 1

There is no x' e IR such that NUF(x') = 1.

 

Hint: For each and every x e IR, x <= 0: NUF(x) = 0 and for each and
every x e IR, x > 0: NUF(x) = aleph_0.

That is blatantly wrong because it would require that ℵo unit fractions
exist between 0 and each and every x > 0,

Which is obviously the case.

Between two unit fractions there are uncountably many x > 0. Therefore your claim is wrong.

If there were a real x with finitely many
UFs less than it, the finitely many larger UFs... couldn't have
infinitely many lesser UFs. Unless you claim finitely many UFs.

There is a smallest UF. This is obviously the case if mathematics is correct: ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 .

 

i.e., the open interval (0, 1].

No, you shifted the quantifiers again.

This is the definition of NUF(x): There are NUF(x) UFs between 0 and x. Not: For given x, there are infinitely many UFs.
If you answer my question with clear mind, then you find: There is no UF smaller than all x > 0 because every UF is an x > 0 itself.
Regards, WM
Regards, WM