Re: how

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.