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

On 10/31/2024 4:13 AM, WM wrote:

On 31.10.2024 01:14, Jim Burns wrote:

On 10/30/2024 3:55 PM, WM wrote:

On 30.10.2024 18:05, Jim Burns wrote:

∀ᴿx > 0: ∀n ∈ ℕ: ⅟⌈n+⅟x⌉ ∈ ⅟ℕ∩(0,x]
>
∀ᴿx > 0: NUF(x) = |ℕ|

>
Is ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 wrong?

>
Neither
  ∀n ∈ ℕ: 1/n - 1/(n+1) > 0
nor
  ∀ᴿx > 0: ∀n ∈ ℕ: ⅟⌈n+⅟x⌉ ∈ ⅟ℕ∩(0,x]
is wrong.

>
But the first formula predicts that
only single unit fractions
are existing on the real line.
How could NUF(x) grow from zero by more than 1?

Is
⎛  ∀ᴿx > 0:
⎜  ∀n ∈ ℕ:
⎝ ⅟⌈n+⅟x⌉ ∈ ⅟ℕ∩(0,x]
wrong?
Is
⎛  ∀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]
wrong?