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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Sat, 2 Nov 2024 22:37:38 +0100
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On 02.11.2024 21:30, Richard Damon wrote:
> On 11/2/24 1:56 PM, WM wrote:
>> On 02.11.2024 17:02, Richard Damon wrote:
>>> On 11/2/24 6:23 AM, WM wrote:
>>
>>>> If NUF(x) has grown to ℵ₀ at x₀, then ℵ₀ unit fractions must be 
>>>> between 0 and x₀. Hence at least ℵ₀ points with ℵ₀ intervals of 
>>>> uncountably many points must be between 0 and x₀. That cannot happen 
>>>> at x₀ = 0.
>>>
>>> Right, NUF(x)is 0 at x - 0, and Aleph_0 at any x > 0, since as you 
>>> said, for any positive finite x, there are Aleph_0 unit fractions 
>>> below it.
>>
>> No, neither did I say so nor is it correct. Only for x > x₀ are ℵ₀ 
>> smaller unit fractions existing.

> So, for what finite number x > 0 is there less than that?

Those numbers are dark.
> 
> There isn't one, as we CAN PROVE that for all x > 0, there exist and 
> infinite number of unit fractions less that x

For every visible number x larger, much larger than x₀ you can prove that.

Regards, WM