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

On 10/8/24 4:31 AM, WM wrote:

On 07.10.2024 13:12, Richard Damon wrote:

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:

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Every point is a finite set.

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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

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The concept of point however does exist. Every single point can be found unless it is dark.
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And every single point that exists can be found, and that includes every point on the "Number Line"

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Find the smallest unit fraction.

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A concept that doesn't exist.
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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.

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No. They nowhere become dense: ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 .
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So, what is the maximum density of unit fractions?

 It is zero because every unit fraction is a point, every gap has uncountable may points.

I guess you don't understand what density means?
The interval from 0.2 (1/5) to 1 has a density of 5 unit fractions per 0.8 units or 50/8 = 6.25 unit fractions per unit distance.
ANY interval starting a zero and moving upwards will have infinite density, because it has an infinite number of points in it.

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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.
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The is no lower limit to the value of 1/n - 1/(n+1) except 0

 This limit is what you can see because you cannot see the reality: Between all points there are giant gaps of uncountable many points.

Right, and that includes between ANY unit fraction and zero, so there can be no smallest unit fraction.
You logic just blows itself up to smithereens due to its contradictions.

 Regards, WM