Re: Does the number of nines increase?

On 7/7/2024 4:04 PM, WM wrote:

Le 07/07/2024 à 19:39, Jim Burns a écrit :

On 7/4/2024 11:16 AM, WM wrote:

Le 04/07/2024 à 17:08, Moebius a écrit :

Am 04.07.2024 um 16:30 schrieb Moebius:

Da gibt's auch nicht zu "diskutieren",

>
Doch, die Frage, wie Du Dir
die Verteilung der ersten Stammbrüche,
also derjenigen, die kleiner als jeder,
den Du angeben kannst, sind, vorstellst, bleibt.
(Off limits for native English speakers.)

>
⎛ ∀j ∈ ℕ₁:
⎜ ∃u ∈ ⅟ℕ: u = ⅟j ∧
⎜ ¬∃v ∈ ⅟ℕ: ⅟j = v ≠ u
⎜

>
The question is only this:
Are there more than one unit fractions in
the point where NUF(x) changes from 0 to more,
or is there only one unit fraction?

The complete distribution of unit fractions,
both darkᵂᴹ and visibleᵂᴹ
is
⎛ ∀B ⊆ ⅟ℕ:  B ≠ {}  ⟹
⎜ ∃v ∈ B ∀u ∈ B: v ≥ u
⎜
⎜ ∀u ∈ ⅟ℕ:  ⅟ℕ\{1} ∋ ⅟(⅟u)⁺¹ < u
⎜
⎝ ∀v ∈ ⅟ℕ\{1}:  ⅟ℕ ∋ ⅟(⅟v)⁻¹ > v
The greatest lower bound β of
{x ∈ ℝ: NUF(x) = ℵ₀}
isn't positive.
⎛ Otherwise,
⎜ if β > 0
⎜ there is a unit fraction u₂ᵦ < 2⋅β such that
⎜ ¼⋅u₂ᵦ < ½⋅β isn't a unit fraction.
⎝ And there isn't such a unit fraction u₂ᵦ
NUF(0) = 0
NUF(x) changes at 0
There is no unit fraction at 0

The question is only this:
Are there more than one unit fractions in
the point where NUF(x) changes from 0 to more,
or is there only one unit fraction?

The third choice: none.