Sujet : Re: How many different unit fractions are lessorequal than all unit fractions?
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.mathDate : 22. Sep 2024, 21:22:05
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <6b50a171-8127-4ce6-9bd3-2dc213638e9b@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
User-Agent : Mozilla Thunderbird
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.
…