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

On 05.09.2024 21:57, Jim Burns wrote:

On 9/5/2024 9:59 AM, WM wrote:

NUF(x) increases from 0 to more.

 ...at 0.

No. NUF(x) counts the unit fractions between 0 and x. Between 0 and 0 there is no unit fraction. NUF(0) = 0.

It cannot increase to 2 or more
before having accepted 1.

 NUFᵈᵉᶠ(x) cannot increase to 2
without having already been ≥ 2

Impossible. NUF(0) = 0. There is a first increase in linear order.

It cannot increase to ℵo without
having accepted 1, 2, 3, ...

 x > 0  ⇒
0 < ... < ⅟⌊4+⅟x⌋ < ⅟⌊3+⅟x⌋ < ⅟⌊2+⅟x⌋ < ⅟⌊1+⅟x⌋ < x

0 is smaller than all that. Therefore there is no increase at 0.
x is larger than all that. Therefore your x is not the least one posiible.
Regards, WM