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

On 10/1/2024 1:29 PM, WM wrote:

On 30.09.2024 21:51, Jim Burns wrote:

On 9/30/2024 2:54 PM, WM wrote:

∀n ∈ ℕ: 1/n - 1/(n+1) > 0
shows that at

>
...no unit.fraction...
>

no point x NUF can increase by
more than one step 1.

>
0 is not a unit.fraction.

 That proves NUF(0) = 0.

>

It is fact with your set too.
I am not responsible.

>
Also,
you are not correct.

>
What is incorrect?

This is incorrect:
🛇⎛ ∀n ∈ ℕ: 1/n - 1/(n+1) > 0  shows that
🛇⎜ at no point x
🛇⎝ NUF can increase by more than one step 1.
∀n ∈ ℕ: 1/n - 1/(n+1) > 0  doesn't show that.
∀n ∈ ℕ: 1/n - 1/(n+1) > 0  shows
∀n ∈ ℕ: 1/n  >  1/(n+1) > 0
which shows
each unit fraction 1/n is not first.

No reason even to care about
mathematical basic truths like
∀n ∈ ℕ: 1/n - 1/(n+1) > 0 ?

>
∀n ∈ ℕ: 1/n  >  1/(n+1) > 0
>
∀n ∈ ℕ: 1/n isn't first unit.fraction.

NUF cannot increase by more than 1 and
cannot start with more than 0 at 0.

NUF(0) = 0
The rest of it, no.
There are
points which are unit fractions
points between unit fractions
points at least some nonzero distance
  from each unit fraction  and
the point 0
You (WM) are mostly considering the first two,
unit fractions and their points.between.
At a point.between unit.fractions,
the set of smaller.unit.fractions doesn't change.
At a unit.fraction,
the set of smaller.unit.fractions changes by 1
⎛ Those sets have a cardinality beyond all
⎜  cardinalities which can change by 1.
⎜ The _set_ changes by 1 element.
⎜ The set's _cardinality_ doesn't change by 1.
⎝ It isn't a finite set, it can't change by 1.
For the third class of points,
before all or after all unit fractions,
plus clearly away from the unit fractions,
NUF = 0 or NUF = ℵ₀ and doesn't change.
Then there's 0.
For every nonzero distance d from 0
there are ℵ₀.many unit.fractions closer than d
∀ᴿd>0:
⎛ u(k) = ⅟⌈k+⅟d⌉
⎜ u: ℕ → (0,d]: one.to.one
⎝ NUF(d) = |u(ℕ)| = |ℕ| = ℵ₀

You should understand that

Do you understand that?

Is that
your concept of
a plan of
a mathematical argument?