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

On 9/21/2024 4:01 PM, WM wrote:

On 20.09.2024 21:51, Jim Burns wrote:

On 9/20/2024 2:24 PM, WM wrote:

On 19.09.2024 20:31, Jim Burns wrote:

On 9/19/2024 6:38 AM, WM wrote:

Either there is a point next to zero
or there is no point next to zero,

>
Consider ABCDEFGHIJKLMNOPQRSTUVWXYZ
>
Q is between A and Z
A and Z are not next to each other.
>
Nothing is between P and Q
P and Q are next to each other.
>
⎛ There is a gap between P and Q

>
No.

>
There is no letter between P and Q
in ABCDEFGHIJKLMNOPQRSTUVWXYZ
>
There is an absence of
letters between P and Q
in ABCDEFGHIJKLMNOPQRSTUVWXYZ

>
No, there is nothing.
An absence requires a space which
could be occupied by the absent.

Either there is a point next to zero
or there is no point next to zero,

For each non.0 point,
there is a space between it and 0
For each non.0 point,
that space is occupied by other points.
There is no point next to 0.
⎛ Assume otherwise.
⎜ Assume δ is next to 0
⎜ 0 < δ ∧ ¬∃ᴿr: 0 < r < δ
⎜
⎜ Consider the unit.fractions ⅟ℕᵈᵉᶠ
⎜ Note that
⎜ ∀⅟k ∈ ⅟ℕᵈᵉᶠ:  ¼⋅⅟k ∈ ⅟ℕᵈᵉᶠ
⎜
⎜ ∀⅟k ∈ ⅟ℕᵈᵉᶠ:
⎜  ¬∃ᴿr: 0 < r < δ ≤ ⅟k
⎜ δ is a lower.bound of ⅟ℕᵈᵉᶠ
⎜
⎜ β the greatest.lower.bound of ⅟ℕᵈᵉᶠ
⎜  might or might not be next to 0  but
⎜ β is greater.equal lower.bound δ
⎜ β = glb.⅟ℕᵈᵉᶠ
⎜ ∀⅟k ∈ ⅟ℕᵈᵉᶠ:
⎜ 0 < δ ≤ β ≤ ⅟k
⎜
⎜ 2⋅β > β
⎜ 2⋅β is not a lower.bound greater.than.greatest
⎜ 2⋅β > ⅟k₂ᵦ
⎜ 2⋅β > ¼⋅⅟k₂ᵦ
⎜ ½.β > ¼⋅⅟k₂ᵦ
⎜ ½.β is not a lower.bound of ⅟ℕᵈᵉᶠ
⎜
⎜ However,
⎜ ½.β < β
⎜ ∀⅟k ∈ ⅟ℕᵈᵉᶠ:
⎜  ½.β < β ≤ ⅟k
⎜ ½.β is a lower bound of ⅟ℕᵈᵉᶠ
⎝ Contradiction.
Therefore
δ isn't next to 0
There is no point next to 0.

Either there is a point next to zero
or there is no point next to zero,

Yes, one is true.
It's the one without darkᵂᴹ numbers.