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

On 10/30/24 7:53 AM, WM wrote:

On 29.10.2024 11:02, FromTheRafters wrote:

WM formulated the question :

On 28.10.2024 21:51, FromTheRafters wrote:

WM explained :

On 28.10.2024 15:07, FromTheRafters wrote:

WM was thinking very hard :

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The function NUF(x) = Number of Unit Fractions between 0 and x starts with 0 at 0.

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Followed by a discontinuity.

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No. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds everywhere.

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Not for n equal to zero.

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I said ∀n ∈ ℕ. 0 is no natural number. NUF(0) = 0.

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That is why your function is a step function with a discontinuity.

 NUF(0) = 0, NUF(1) = ℵo. Therefore NUF must grow but cannot grow by more than 1 at any point x of the real axis.
 Regards, WM
 

Because it grows BETWEEN points on the real axis from 0 to ℵo
Your problem is there isn't a "first" x > 0 above 0 to see it increase at, so it can just jump to ℵo.
Your logic has just blown up your system to smithereens.