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

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?
Regards, WM