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

Am Fri, 06 Sep 2024 22:52:37 +0200 schrieb WM:

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.

Which are yet to be defined.

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

Betwee 0 and every epsilon you can define.

Enough for me.

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.

First, not last.

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.

So what?

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.

What about the negative of Dirac’s delta?

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?

You in particular should know the reals are dense. There is no
„immediately after”.

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.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.