Re: how

On 5/26/2024 2:53 PM, WM wrote:

Le 23/05/2024 à 21:52, Jim Burns a écrit :

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

I have shown the way: Dark numbers.

>
Darkᵂᴹ numbers in ℚ and between splits of ℚ
which are between 0 and all unit fractions
do not exist, neither darklyᵂᴹ nor visiblyᵂᴹ

>
What is closer to zero,
a unit fraction or a not unit fraction?

Which is larger,
an even number or an odd number?
Neither.

We can know that they don't exist by starting with
that description and then making not.first.false
claims until we get to a contradiction.

 The contradiction is
∀x ∈ (0, 1]: NUF(x) = ℵo
because the unit fractions are x ∈ (0, 1].

For any x > 0
there are more.than.any.k<ℵ₀ unit.fractions < x
among which are  ⅟⌊(1+⅟x)⌋  ...  ⅟⌊(k+1+⅟x)⌋

They cannot sit at a single point x,
hence the statememt is false.
>

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

>
That implies that
there is a unit fractions smaller than
all other x > 0.

Show your work.
And, by "Show your work", I do not mean
"Repeat your unsupported claim endlessly".
Explain how both your claim is true and,
for each x > 0,  ⅟⌊(1+⅟x)⌋ < x exists,  and
for each ⅟n,  ⅟(n+π) < ⅟n exists.

and even in accordance with
For any unit fraction there are ℵ₀ smaller real x > 0.

>
Also true:
For any x > 0 there are ℵ₀ smaller unit fractions.

>
Impossible because
the unit fractions cannot be smaller than themselves.

For any x > 0
there are more.than.any.k<ℵ₀ unit.fractions < x
among which are  ⅟⌊(1+⅟x)⌋  ...  ⅟⌊(k+1+⅟x)⌋
none of which are smaller than themselves.