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

On 06.09.2024 18:41, Jim Burns wrote:

On 9/6/2024 8:12 AM, WM wrote:

On 05.09.2024 21:57, Jim Burns wrote:

On 9/5/2024 9:59 AM, WM wrote:

 

NUF(x) increases from 0 to more.

>
...at 0.

>
No.
NUF(x) counts the unit fractions between 0 and x.
Between 0 and 0 there is no unit fraction.
NUF(0) = 0.

 Between 0 and x
there are more than 0 unit fractions

Between 0 and your defined x or epsilon, not between 0 and every possible x.

Between 0 and x
there are more.than.any.k<ℵ₀ unit fractions.

Betwee 0 and every epsilon you can define.

NUF(0) = 0.
There is a first increase in linear order.

 ∀ᴿx ∈ (-1,0]:  ⌈x⌉ = 0
∀ᴿx ∈ (0,1]:  ⌈x⌉ = 1
 Is there a first ε at which 5⋅⌈ε⌉ = 5 ?

ε = 1. But that is not related to the topic.

0 is smaller than all that.
Therefore there is no increase at 0.

 Increases and decreases involve nearby points
from which increases and decreases increase and decrease.

But here we have always 2^ℵo points where the function is constant before it increases by 1.

Can you answer
whether f(x) increases or decreases at 0?

It is constant 0.

 No, you can't answer without information about
nearby points.

I have information about nearby points on the negative axis. Only nearby points less than x are relevant for judging about an increase in x. Nearby points larger than x are irrelevant.

 

x is larger than all that.

 "All that" are more.than.any.k<ℵ₀ unit.fractions.
NUF(x) = ℵ₀
 

Therefore your x is not the least one posiible.

 Yes.
x is NOT the first point > 0 after
more.than.any.k<ℵ₀ unit.fractions.

But there is an x immediately after 0. What else should be there?

 Generalize.
What do we know about x ?
x > 0
What else?

We do not know anything. But we know that all unit fractions differ from each other. That is sufficient to know that NUF(x) increases by 1 only at any unit fraction.
Regards, WM