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

On 9/22/2024 2:37 PM, WM wrote:

On 22.09.2024 19:44, Jim Burns wrote:

There is no point next to 0.

There is a smallest unit fraction
because
there are no unit fractions without a first one
when counting from zero.

[1/2]  ∀⅟k ∈ ⅟ℕᵈᵉᶠ:  ⅟ℕᵈᵉᶠ ∋ ¼⋅⅟k < ⅟k
Each visibleᵂᴹ unit.fraction ⅟k has
a counter.example ¼⋅⅟k to its being smallest.
[2/2]  ¬∃ᴿδ > 0: ¬( δ >ₑₓᵢₛₜₛ ⅟ℕᵈᵉᶠ )
No positive point, unit.fraction or otherwise,
is NOT undercut by some visible unit fraction.
A point undercut by a visibleᵂᴹ unit fraction
is not the smallest unit fraction.
⎛ Assume otherwise.
⎜ Assume
⎜ δ > 0  ∧  ¬( δ >ₑₓᵢₛₜₛ ⅟ℕᵈᵉᶠ )
⎜
⎜ β = glb.⅟ℕᵈᵉᶠ ≥ δ > 0
⎜
⎜ ½.β < β
⎜ ¬( ½.β >ₑₓᵢₛₜₛ ⅟ℕᵈᵉᶠ )
⎜
⎜ 2.β > β
⎜ 2.β >ₑₓᵢₛₜₛ ⅟ℕᵈᵉᶠ
⎜
⎜ However,
⎜ 2.β >ₑₓᵢₛₜₛ ⅟ℕᵈᵉᶠ
⎜ 2.β > ⅟k
⎜ ½.β > ¼⋅⅟k
⎜ ½.β >ₑₓᵢₛₜₛ ⅟ℕᵈᵉᶠ
⎜
⎜ ½.β >ₑₓᵢₛₜₛ ⅟ℕᵈᵉᶠ  ∧  ¬( ½.β >ₑₓᵢₛₜₛ ⅟ℕᵈᵉᶠ )
⎝ Contradiction.
Therefore,
¬∃ᴿδ > 0: ¬( δ >ₑₓᵢₛₜₛ ⅟ℕᵈᵉᶠ )
Visibleᵂᴹ or darkᵂᴹ,
there is no smallest unit fraction.