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

On 9/9/2024 2:44 PM, WM wrote:

On 09.09.2024 17:15, joes wrote:

I still don’t understand.
You can choose any size of interval and
slide it around to include
an arbitrary number of unit fractions.

>
ℵo unit fractions exist invariably

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.

and require a minimum length d.

No.
There is a greatest.lower.bound 0
but there isn't  a _minimum_
ℵ₀.many unit.fractions do not
  fit in a 0.length interval.
For each d > 0
ℵ₀.many unit.fractions fit in (0,d)
k ⟼ ⅟⌊k+⅟d⌋
and
for ½⋅d
ℵ₀.many unit.fractions fit in (0,½⋅d)
k ⟼ ⅟⌊k+2⋅⅟d⌋
-- which aren't the same as ⅟⌊k+⅟d⌋ --
so
each d > 0 is not.minimum.