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

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

Le 04/09/2024 à 23:26, Jim Burns a écrit :

On 9/4/2024 4:23 PM, WM wrote:

But all different unit fractions are different,
i.e., they sit at different positive x.

>
Yes,
each two sit at two points,
because ∀n ∈ ℕ: ⅟n-⅟(n+1) > 0
>

Therefore only one can sit closest to zero.

>
No,
each sits not.closest to zero,
because ∀n ∈ ℕ: ⅟n-⅟(n+1) > 0

>
NUF(x) increases from 0 to more.

...at 0.
∀n ∈ ℕ: ⅟n ≠ 0

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

NUFᵈᵉᶠ(x) cannot increase to 2
without having already been ≥ 2
0 < ... < ⅟(n+2) < ⅟(n+1) < ⅟n
NUF(x) ≥ NUFᵈᵉᶠ(x)
x > 0  ⇒  NUF(x) ≥ NUFᵈᵉᶠ(x) ≥ 2
x ≤ 0  ⇒  NUF(x) = 0
x ⪋ 0  ⇒  NUF(x) ≠ 1

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