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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: how
Date: Fri, 17 May 2024 01:00:54 +0200
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Am 16.05.2024 um 20:56 schrieb Jim Burns:
> On 5/15/2024 9:12 AM, WM wrote:
>> Le 14/05/2024 à 15:35, Moebius a écrit :
>>> Am 14.05.2024 um 13:13 schrieb WM:
> 
>>>> Where can the first unit fractions exist
>>>> on the real line
>>>
>>> Nowhere.
>>> Since there is no such unit fraction,
>>> it can't be anywhere.
>>
>> Every subset of the real line has
>> a first element.
> 
> No.
> 
> | Assume ⅟ℕ∩(0,1] has first element ⅟G
> |
> | 0 < ½⋅⅟G < ⅟G < 2⋅⅟G
> | There IS a unit.fraction ⅟k < 2⋅⅟G
> | There is NOT a unit.fraction < ½⋅⅟G
> |
> | ⅟k < 2⋅⅟G exists
> | (⅟k)/4 < (2⋅⅟G)/4
> | ⅟(4⋅k) < ½⋅⅟G
> | There IS a unit.fraction ⅟(4⋅k) < ½⋅⅟G
> | Contradiction.
> 
> Therefore,
> ⅟ℕ∩(0,1] does NOT have a first element.

How about?

Let S = {x e IR : 0 < x}. Then S is a (nomempty) subset of IR.
Assume there is a first/smallest element in S. Let r be this element. 
Then r e IR and 0 < r and hence r/2 e IR and 0 < r/2. Hence r/2 in S. 
But r/2 < r. Contradiction! Hence S does not have a first/smallest 
Element. This disproves WM's claim. [ ]

It seems that WM likes to assert false statements.