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

On 11.09.2024 20:11, Jim Burns wrote:

On 9/11/2024 10:48 AM, WM wrote:

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.

 ⅟⌊k+⅟x⌋ ⟷ ⅟⌊k+⅟x′⌋
is a bijection between different visibleᵂᴹ unit.fractions
⅟⌊k+⅟x⌋ ≠ ⅟⌊k+⅟x′⌋

So it appears, but that does not contradict what I said.

0, ..., 1/100, x', 1/99, ..., 1/75, x, 1/74, ..., 1/1.

 0 is not a unit.fraction.

No, but I cannot give the smallest unit fraction.
 From it to 1/100 all unit fractions are the same in (0, x) and (0, x').

 ..., 1/100, x', 1/99, ..., 1/75, x, 1/74, ..., 1/1.
holds only visibleᵂᴹ unit fractions

The first ℵo unit fractions are dark.
This example shows that you were mistaken. All intervals (0, x) with visible x have a set of ℵo unit fractions between (0 and x) in common.
Regards, WM