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

On 9/2/2024 1:07 PM, WM wrote:

How many different unit fractions are
lessorequal than all unit fractions?
The correct answer is: one unit fraction.

Another answer is that
no unit fraction is
lessorequal than all unit fractions.

Yes:
for each ⅟j ∈ ⅟ℕᴰᴱꟳ
there is next.smaller ⅟j′ ∈ ⅟ℕᴰᴱꟳ:
⎛ ⅟j′ = ⅟(j+1) < ⅟j
⎝ ¬∃⅟hₓ ∈ ⅟ℕᴰᴱꟳ: ⅟j′ < ⅟hₓ < ⅟j
Also:
for each non.max.⅟ℕᴰᴱꟳ ⅟j ∈ ⅟ℕᴰᴱꟳ
there is next.larger ⅟j″ ∈ ⅟ℕᴰᴱꟳ:
⎛ ⅟j″ = ⅟(j-1) > ⅟j
⎝ ¬∃⅟hₓ ∈ ⅟ℕᴰᴱꟳ: ⅟j″ > ⅟hₓ > ⅟j
⇐  ∃⅟i ∈ ⅟ℕᴰᴱꟳ: ⅟i > ⅟j
for each non.{} S ⊆ ⅟ℕᴰᴱꟳ
there is max.S ⅟j ∈ S:
S ∋ ⅟j ≥ᵉᵃᶜʰ S
for each bounded.in.⅟ℕᴰᴱꟳ non.{} S ⊆ ⅟ℕᴰᴱꟳ
there is min.S ⅟j ∈ S:
S ∋ ⅟j ≤ᵉᵃᶜʰ S
⇐  ∃⅟k ∈ ⅟ℕᴰᴱꟳ: ⅟k ≤ᵉᵃᶜʰ S
0 ∉ ⅟ℕᴰᴱꟳ

Another answer is that
no unit fraction is
lessorequal than all unit fractions.
That means the function NUF(x)
Number of UnitFractions between 0 and x > 0
with NUF(0) = 0 will never increase but stay at 0.

That means that
the unit.fractions between 0 and x > 0
will never be fewer than ℵ₀.many
well.ordered, stepping.up.and.down.except.max
visibleᵂᴹ unit.fractions.
⎛ In a finite non.{} ordered set,
⎝ each subset is 2.ended.
That means that
the 1.ended unit.fractions between 0 and x > 0
are not finitely.many.

That means the function NUF(x)
Number of UnitFractions between 0 and x > 0
with NUF(0) = 0 will never increase but stay at 0.
There are no unit fractions existing at all.

x > ⅟⌊1+⅟x⌋ > 0
The unit.fractions are 1.ended.
There is no first unit.fraction.

Therefore
there is only the one correct answer given above.

Logicᵂᴹ which leads to the claim that
⎛ half the greatest.lower.bound of visibleᵂᴹ unit.fractions
⎜ is undercut by visibleᵂᴹ unit.fractions and
⎝ is NOT undercut by visibleᵂᴹ unit.fractions
is broken logicᵂᴹ.
Not all sets are finite.
Not all sets have each non.{} subset 2.ended.