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

On 10/31/24 1:28 PM, WM wrote:

On 31.10.2024 12:36, Richard Damon wrote:

On 10/30/24 7:53 AM, WM wrote:

 

NUF(0) = 0, NUF(1) = ℵo. Therefore NUF must grow but cannot grow by more than 1 at any point x of the real axis.
>

Because it grows BETWEEN points on the real axis from 0 to ℵo.

 I investigate its behaviour at single points.

>
Your problem is there isn't a "first" x > 0 above 0 to see it increase at,

 We do not see dark numbers but I can prove their existence by
∀n ∈ ℕ: 1/n - 1/(n+1) > 0.

And we can show that those dark numbers were just ordinary Natural Numbers that you just couldn't handle.

 

 so it can just jump to ℵo.

But it cannot do so at a single point.

But there isn't a "single point" that it needs to jump at.
We agree that at 0, NUF(x) is 0.
There is no "next" point to be that "single point" that it would need to jump at.
We cross an unbounded border where we jump to the new value.

 Regards, WM