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

On 11/1/2024 1:24 PM, WM wrote:

On 01.11.2024 14:21, Jim Burns wrote:

⎛  ∀ᴿx > 0:
⎜  ∀n ∈ ℕ:
⎜ x > 0
⎜ ⅟x > 0
⎜ n+⅟x ≥ ⅟x > 0
⎜ ⌈n+⅟x⌉ ≥ ⅟x > 0
⎜ ⌈n+⅟x⌉ ∈ ℕ
⎜ ⅟⌈n+⅟x⌉ ∈ ⅟ℕ
⎜ x⋅⌈n+⅟x⌉ ≥ x⋅⅟x > 0
⎜ x⋅⌈n+⅟x⌉⋅⅟⌈n+⅟x⌉ ≥ 1⋅⅟⌈n+⅟x⌉ > 0
⎜ x ≥ ⅟⌈n+⅟x⌉ > 0
⎜ ⅟⌈n+⅟x⌉ ∈ (0,x]  ∧  ⅟⌈n+⅟x⌉ ∈ ⅟ℕ
⎝ ⅟⌈n+⅟x⌉ ∈ ⅟ℕ∩(0,x]
 ⎛  ∀n ∈ ℕ⁺:
⎜ n > 0
⎜ n+1 > n > 0
⎜ ⅟n⋅(n+1)⋅⅟(n+1) > ⅟n⋅n⋅⅟(n+1)
⎜ ⅟n > ⅟(n+1)
⎜ ⅟n - ⅟(n+1) > ⅟(n+1) - ⅟(n+1)
⎝ ⅟n - ⅟(n+1) > 0

Will you (WM) also be ignoring
the work which shows
NUF(x) grows at 0 to ℵ₀ ?

>
Of course.

If NUF(x) has grown to ℵ₀ at x₀,
then ℵ₀ unit fractions must be between 0 and x₀.

Yes, and ℵ₀.many are there,
because x₀ ≥  ⅟⌈n+⅟x₀⌉ > 0
NUF(0) = 0  ∧
  ∀ᴿx₀ > 0:
NUF(x₀)=ℵ₀  ⇐  ∀n∈ℕ: (0,x₀] ∋ ⅟⌈n+⅟x₀⌉ ∈ ⅟ℕ

Hence
at least ℵ₀ points with
ℵ₀ intervals of uncountably many points
must be between 0 and x₀.

That cannot happen at x₀ = 0.

No.
There is no x₀ > 0 which,
  between it and 0,
   that does not happen.
NUF(0) = 0  ∧
  ¬∃ᴿx₀ > 0:
NUF(x₀)<ℵ₀  ∧  ∀n∈ℕ: (0,x₀] ∋ ⅟⌈n+⅟x₀⌉ ∈ ⅟ℕ

Therefore
not even a fool could support your claim.

E pur si muove:
x₀ ≥  ⅟⌈n+⅟x₀⌉ > 0