Path: ...!weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: Ben Bacarisse <ben@bsb.me.uk>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit fractions?
Date: Mon, 16 Sep 2024 23:02:16 +0100
Organization: A noiseless patient Spider
Lines: 35
Message-ID: <87v7yvfeaf.fsf@bsb.me.uk>
References: <vb4rde$22fb4$2@solani.org> <vbq4ve$31fu6$10@dont-email.me>
	<fd09e9afa6b0c3041b90c5d788681bb2c92f9d2e@i2pn2.org>
	<vbs9v8$3l368$3@dont-email.me>
	<405557f7289631d63264c712d137244c940b9926@i2pn2.org>
	<vbsroa$3mvi7$2@dont-email.me> <vbt0fs$3pr1d$1@dont-email.me>
	<vbt15n$3qapk$1@dont-email.me> <btbO__HYVIMPfoOcmXp4_whV8-8@jntp>
	<vbums6$8kdn$1@dont-email.me> <vbv9i7$bpjh$2@dont-email.me>
	<13c08e96ad635f8142b38d89863a80caf17a32a8@i2pn2.org>
	<vc1mfe$u3ec$2@dont-email.me>
	<4faa63d0ff8c163f01a38736aeb5732184218a29@i2pn2.org>
	<vc1uu8$u3ec$9@dont-email.me> <vc2gfb$130uk$1@dont-email.me>
	<vc44uu$1gc40$1@dont-email.me>
	<c94d6140f000f75c5e95e1acc785ebff9894a18b@i2pn2.org>
	<vc58cf$1o4dp$3@dont-email.me> <87jzfchnxc.fsf@bsb.me.uk>
	<vc7d5u$2atht$2@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain
Injection-Date: Tue, 17 Sep 2024 00:02:17 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="23cfe63829707f788be76e26d9d06d0a";
	logging-data="3229227"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1+pV2IdUIqMvJBfxweuB4Ej/IUdwe35u7A="
User-Agent: Gnus/5.13 (Gnus v5.13)
Cancel-Lock: sha1:f2x+rwIdE9VLmux4gUu8JhU0Wv4=
	sha1:f3PenCyuICcN9SBbasl810NptmQ=
X-BSB-Auth: 1.7aaf8cc669dd6beee84c.20240916230216BST.87v7yvfeaf.fsf@bsb.me.uk
Bytes: 3373

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> On 9/15/2024 9:38 AM, Ben Bacarisse wrote:
>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>> 
>>> On 9/14/2024 11:35 AM, joes wrote:
>>>> Am Sat, 14 Sep 2024 16:01:02 +0200 schrieb WM:
>>>>> On 14.09.2024 01:05, FromTheRafters wrote:
>>>>>> WM explained :
>>>>>
>>>>>>> No, that is your big mistake. In the interval [0, 1] there is a point
>>>>>>> next to 0 and a point next to 1, and infinitely many are beteen them.
>>>>>> Define 'next' in this context.
>>>>> Two points are next to each other means that no point is between them.
>>>> Which is the case for no two (different) reals.
>>>
>>> Two points on the real line that are different from one another have
>>> infinite points between them, and so on and so forth. :^)
>> It might be worth pointing out that any non-trivial interval [a, b] on
>> the real line (i.e. with b > a) contains an uncountable number of
>> points.  Constructing the mid point and the quarter points and so on
>> only shows a countably infinite number of internal points, but giving a
>> bijection between [a, b] and [0, 1] shows that they have the same
>> cardinality.
>
> Agreed. Afaict, one way to cover all the points is to draw a solid line
> between two different points p0 and p1 where p0 does not equal p1. We can
> say the line covers them all? Fair enough? Put two different points on a
> piece of paper and draw a line from p0 to p1. That line contains infinitely
> dense points.

Too much physics for me to comment.

-- 
Ben.