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

On 9/13/2024 11:43 AM, WM wrote:

On 12.09.2024 20:16, Richard Damon wrote:

On 9/12/24 1:44 PM, WM wrote:

Therefore the end of an open interval consist of points.

>
The "end" of the open interval isn't in the interval,
that is why it is called "open".

>
The end of the interval is a point of the interval.
It is called open because next to a definable point
there are always dark points.

An open set holds none of its boundary.
A closed set holds all of its boundary.
If intervals holding endpoints are open,
then single.point interval.intersections are open
and arbitrary unions of single.point.sets are open
and all point.sets are open.
"All point.sets open" is the devolution of
"Obviously, Bob cannot disappear".
We are finite.
Infinity is foreign.

There is no "first" point of that interval,
just an infinite set of points

>
Yes, dark points.

Points in ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]] have no
🛇 first in (0,1]
Points in ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]] have no
🛇 positive lower bound of {⅟n: n ∈ ℕ[⟨1,…,n⟩]}
⎛ r′ ∈ ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]] locates
⎜ the open.split S′ ⊆ ℚ[ℤ[ℕ[⟨0,…,n⟩]]]
⎜ {} ≠ S′ ᵉᵃᶜʰ< r′ ≤ᵉᵃᶜʰ (ℚ[ℤ[ℕ[⟨0,…,n⟩]]] \ S′) ≠ {}
⎜
⎜ q′ ∈ ℚ[ℤ[ℕ[⟨0,…,n⟩]]] is the quotient z′/z"
⎜ z′,z" ∈ ℤ[ℕ[⟨0,…,n⟩]] and z"≠0
⎜
⎜ z′ ∈ ℤ[ℕ[⟨0,…,n⟩]] is the difference n′-n"
⎜ n′,n" ∈ ℕ[⟨0,…,n⟩]
⎜
⎜ n′ ∈ ℕ[⟨0,…,n⟩] ends FISON ⟨0,…,n′⟩
⎜
⎜ ⟨0,…,n′⟩ is well.ordered (minimummed non.empties)
⎜ for k ∈ ⟨1,…,n′⟩:  ⟨0,…,n′-1⟩ ∋ k-1
⎝ for j ∈ ⟨0,…,n′⟩:  ⟨1,…,n′+1⟩ ∋ j+1

just an infinite set
of points approaching that point.

>
Points do not approach. They are fixed.

Sets approach. Sets aren't points.
⎛ Assume
⎜ ∃δ: 0 < δ ≤ᵉᵃᶜʰ ⅟ℕ[⟨1,…,n⟩]
⎜
⎜ β = glb.⅟ℕ[⟨1,…,n⟩]
⎜ α < β  ⇒  α ≤ᵉᵃᶜʰ ⅟ℕ[⟨1,…,n⟩]
⎜ γ > β  ⇒  ¬(γ ≤ᵉᵃᶜʰ ⅟ℕ[⟨1,…,n⟩])
⎜ γ > β  ⇒  γ >ₑₓᵢₛₜₛ ⅟ℕ[⟨1,…,n⟩]
⎜
⎜ 2⋅β > β
⎜ 2⋅β >ₑₓᵢₛₜₛ ⅟ℕ[⟨1,…,n⟩]
⎜ 2⋅β > ⅟k ∈ ⅟ℕ[⟨1,…,n⟩]
⎜ ½⋅β > ¼⋅⅟k ∈ ⅟ℕ[⟨1,…,n⟩]  [!]
⎜ ½⋅β >ₑₓᵢₛₜₛ ⅟ℕ[⟨1,…,n⟩]
⎜ ¬(½⋅β ≤ᵉᵃᶜʰ ⅟ℕ[⟨1,…,n⟩])
⎜
⎜ However,
⎜ ½⋅β < β
⎜ ½⋅β ≤ᵉᵃᶜʰ ⅟ℕ[⟨1,…,n⟩]
⎝ Contradiction.
¬∃δ: 0 < δ ≤ᵉᵃᶜʰ ⅟ℕ[⟨1,…,n⟩]
Nothing exists between 0 and ⅟ℕ[⟨1,…,n⟩]
⅟ℕ[⟨1,…,n⟩] approaches 0.