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

On 9/6/24 8:12 AM, WM wrote:

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.

But it CAN'T for the values of x that are finite, as for ANY finite number, there is an infinite number of unit factions below it, so it can't ever have a finite value for any finite x.
So, either NUF(x) counts some sub-finite "unit fractions" (that are not the reciprocals of Natural Numbers) or it just doesn't exist.

 

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.

Which can't be the reciprical of a Natural Numbers (since there will always be a smaller number) so it must be counting some sub-finite numbers as unit fractions or it just doesn't exist.

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.

So it can't increment finitely at recipricals of Natural Numbers so either it is counting some sub-finite values too, or it just doesn't exist.

 Regards, WM