Re: How many different unit fractions are lessorequal than all unit fractions?

On 10.09.2024 23:12, Jim Burns wrote:

On 9/10/2024 12:16 PM, WM wrote:

On 09.09.2024 21:43, Jim Burns wrote:

 

The unit fractions
  before x > 0 and before x′ > 0
vary.
For both x and x′, they are ℵ₀.many,
but not the same ℵ₀.many.

>
The same ℵ₀ many are existing in both cases,

 No.
They are not the same for ⌊⅟x⌋ < ⌊⅟x′⌋

The unit fractions are fixed at the real line. If you cut some more or some less by x than by x' is irrelevant for the fact that the same ℵo unit fractions remain in both cases.
0, ..., 1/100, x', 1/99, ..., 1/75, x, 1/74, ..., 1/1.

_Minimum distances of countably.many.points does not exist_
 

Every uncountable set has a countable subset, in fact uncoubtably many.
Therefore (0, d) c (0, x). (0, d) is coubtable and contains at most one unit fraction.
Regards, WM