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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: density of the reals
Date: Wed, 11 Sep 2024 16:33:14 +0200
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On 11.09.2024 08:05, joes wrote:
> Am Tue, 10 Sep 2024 21:13:57 +0200 schrieb WM:
>> On 10.09.2024 20:38, Chris M. Thomasson wrote:
>>> It seems that any non-zero gap can have unit fractions small enough to
>>> fit in it...
>> A gap of countably many points is not sufficient but is a subgap of
>> every gap between unit fractions.
> Why is it not sufficient? 

Because between any two unit fractions there are uncountably many points.

> Since when are the reals countable?

Every uncountable set has a countable subset, in fact uncountably many. 
The countable set (0, a) is a subinterval of the uncountable set (0, d) 
comprising ℵo unit fractions. No x of (0, a) can have smaller unit 
fractions. Well, perhaps one.

Regards, WM
>