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

WM explained :

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.

 I meant 'step-wise' function.
 

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.

 The number of unit fractions less than x is always aleph_zero for positive x. Nothing about growing them in some sort of unit fraction backyard garden involved.