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

On 10/7/24 4:56 AM, WM wrote:

On 06.10.2024 19:11, Richard Damon wrote:

On 10/6/24 9:42 AM, WM wrote:

On 06.10.2024 04:51, Richard Damon wrote:

On 10/5/24 2:58 PM, WM wrote:

>

Every point is a finite set.

>

The fact that we can keep doing that indefinitely, and never reach a point we can't continue, is proof that the concept of "next point" doesn't exist

>
The concept of point however does exist. Every single point can be found unless it is dark.
>

And every single point that exists can be found, and that includes every point on the "Number Line"

 Find the smallest unit fraction.

A concept that doesn't exist.

Except the "Unit Fractions" or the "Natural Numbers" those have gaps between the elements, with an "accumulation point" at 0 where the Unit Fractions become dense there.

 No. They nowhere become dense: ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 .
 

So, what is the maximum density of unit fractions?
The fact that 1/n - 1/(n+1) > 0 just means that 1/n > 1/(n+1) and thus there is always a unit fraction smaller than the one you have.
The is no lower limit to the value of 1/n - 1/(n+1) except 0, so by going far enough we can find two unit fractions as close to each other as we want, and there will always be an infinite set smaller than that.
Dense doesn't mean points on top of each other, but that there is always a point in between.
The fact that you don't understand this, is why you keep getting the wrong answers because you brain has been exploded to smithereens by the contradictions in your logic.

Regards, WM