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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Thu, 31 Oct 2024 21:18:43 +0100
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On 31.10.2024 19:34, Jim Burns wrote:
> On 10/31/2024 1:41 PM, WM wrote:
>> On 31.10.2024 13:22, Jim Burns wrote:
>>> On 10/31/2024 4:13 AM, WM wrote:
> 
>>>>> 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?
>>
>> One of two contradicting formulas must be dropped.
> 
> Which (inside.quantifiers) formula is
> the last which you accept with all prior formulas?

Which one requires that NUF(x) can grow at an x ∈ ℝ by more than 1?

Regards, WM