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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 18:56:51 +0100
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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.

Regards, WM