Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: =?UTF-8?Q?Re=3A_There_is_a_first/smallest_integer_=28in_M=C3=BCcken?= =?UTF-8?Q?land=29?= Date: Wed, 17 Jul 2024 12:51:21 -0700 Organization: A noiseless patient Spider Lines: 31 Message-ID: References: <7mXj2D8kEhAscu3HLTqTUKsaj18@jntp> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Wed, 17 Jul 2024 21:51:23 +0200 (CEST) Injection-Info: dont-email.me; posting-host="3f8b7cebb49a7a47651d12f9ce849b53"; logging-data="2088092"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19L7p8c2JG71Y+PvwVReOWAYUFMDT/XNnA=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:mg+vNbbxmhOpKafSWYb6WvUT9J0= Content-Language: en-US In-Reply-To: Bytes: 2411 On 7/17/2024 10:17 AM, WM wrote: > Le 17/07/2024 à 19:01, joes a écrit : >> Am Wed, 17 Jul 2024 15:08:30 +0000 schrieb WM: >>> Le 17/07/2024 à 16:56, Moebius a écrit : >>>> Am 17.07.2024 um 16:43 schrieb WM: >>> >>>>> Can you explain how NUF(x) can [jump] from 0 [at x = 0] to [aleph_0] >>>>> [at any] >>>>> point x [> 0] although all unit fractions are separated by finite >>>>> distances [...] >>>> >>>> Yes, of course: For each and every x e IR, x > 0 there are >>>> countably-infinitely many unit fractions which are <= x. (Hint: No >>>> first one.) >>> >>> Thema verfehlt. The question is: How does NUF(x) increase from 0 to >>> more? There is a point where NUF is 0 and then it increases. How? >> The same as the sign function. > > No, ℵo finite intervals do not fit between [0, 1] and (0, 1]. The sign > function fits. > >> There simply is no such "point", as >> there is no least positive number. The distances between unit >> fractions get infinitely small. > > They remain finite in every case. There are infinitely many of them, and none of them equals zero... 0/1 is not a unit fraction! Damn it. :^)