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

On 07.09.2024 01:51, Jim Burns wrote:

On 9/6/2024 4:52 PM, WM wrote:

>
Between 0 and your defined x or epsilon,
not between 0 and every possible x.

 If x > 0  then there is ⅟⌊1+⅟x⌋  between 0 and x

Then identical unit fractions differ.
They are identical because NUF(x) counts them at the same x, but they differ because NUF(x) counts more than 1 at this x.

 A quantifier shift is not reliable.

That is no quantifier shift but simplest logic.

 

Between 0 and x
there are more.than.any.k<ℵ₀ unit fractions.

>
Between 0 and every epsilon you can define.

 Between 0 and any epsilon satisfying my description.
I haven't made a claim for other epsilons.

Fine. There I agree. Every of your epsilons has ℵo smaller unit fractions in (0, eps). This proves dark numbers.
Regards, WM