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

On 9/4/2024 4:23 PM, WM wrote:

On 04.09.2024 20:08, Jim Burns wrote:

On 9/2/2024 1:07 PM, WM wrote:

How many different unit fractions are
lessorequal than all unit fractions?
The correct answer is: one unit fraction.

>

Another answer is that
no unit fraction is
lessorequal than all unit fractions.

>
Then NUF(x) will never leave the value 0.
>

Not all sets are finite.

>
But all different unit fractions are different,
i.e., they sit at different positive x.

Around each of ℵ₀.many visibleᵂᴹ ⅟k,
mark a gap of length g/2ᵏ
Gap length g/2ᵏ  decreases exponentially faster than
gap distance ⅟k-⅟(k+1)
If gaps do not overlap initially,
then they do not overlap ever.
⎛ Gₖ = g/2+g/4+g/8+...+g/2ᵏ
⎜
⎜ ½⋅(g+Gₖ)  =
⎜ ½⋅(g+g/2+g/4+g/8+...+g/2ᵏ)  =
⎜ g/2+g/4+g/8+...+g/2ᵏ+g/2ᵏ⁺¹  =
⎜ Gₖ+g/2ᵏ⁺¹
⎜
⎜ ½⋅(g+Gₖ)  =  Gₖ+g/2ᵏ⁺¹
⎝ Gₖ  =  g-g/2ᵏ  <  g
For visibleᵂᴹ unit.fraction ⅟j, let g = ½⋅⅟j²
ℵ₀.many visibleᵂᴹ unit.fractions ⅟ℕᴰᴱꟳ and the gaps g/2ᵏ
fit between 0 and ⅟j

But all different unit fractions are different,
i.e., they sit at different positive x.

Yes,
each two sit at two points,
because ∀n ∈ ℕ: ⅟n-⅟(n+1) > 0

Therefore only one can sit closest to zero.

No,
each sits not.closest to zero,
because ∀n ∈ ℕ: ⅟n-⅟(n+1) > 0