Re: how

On 5/26/2024 3:15 PM, WM wrote:

Le 25/05/2024 à 19:23, Jim Burns a écrit :

On 5/23/2024 8:10 AM, WM wrote:

There is no unit fraction smaller than all x > 0, [A]

>
Also true:
There is no x > 0 smaller than all unit fractions.  [B]

Note that
points on the real axis are fixed and
can be subdivided into two sets, namely
the set of unit fractions and
the set of positive non-unit fractions.

Agreed.

If A is true, then there is
a positive non-unit fraction smaller than
all unit fractions.

No.
| Assume that claim is correct.
| Assume that x > 0 is smaller than
| all unit.fractions.
|
| However,
| ⅟⌊(1+⅟x)⌋ is a unit.fraction < x
| Contradiction.
That claim is incorrect.

If B is true, then there is
a unit fraction smaller than
all positive non-unit fraction.

No.
| Assume that claim is correct.
| Assume that ⅟n is smaller than
| all positive non.unit.fractions.
|
| However,
| ⅟(n+π) is a non.unit.fraction < ⅟n
| Contradiction.
That claim is incorrect.

There is no third alternative.

Consider ⅟⌊(1+⅟x)⌋ and ⅟(n+π)
There is neither
a unit.fraction before all non.unit.fractions > 0
nor
a non.unit.fractions > 0 before all unit.fractions.
Also,
there is neither
an even number after all odd numbers
nor
an odd number after all even numbers.

But there is no decision possible.

The decision is that not all subsets
of unit.fractions or of non.unit.fractions
have two ends.