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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Gaps... ;^)
Date: Tue, 10 Sep 2024 21:23:54 +0200
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Am 10.09.2024 um 20:30 schrieb Chris M. Thomasson:
> On 9/9/2024 5:28 PM, Moebius wrote:
>> Am 10.09.2024 um 00:59 schrieb Chris M. Thomasson:
>>
>>> Between zero and any positive x there is a unit fraction small enough 
>>> to fit in the ["]gap["].
>>
>> Right. This follows from the so called "Archimedean property" of the 
>> reals. From this property we get:
>>
>> For all x e IR, x > 0, there is an n e IN such that 1/n < x.
>>
>> See: https://en.wikipedia.org/wiki/Archimedean_property
>>
>> Of course, from this we get that there are infinitely many unit 
>> fractions smaller than x, say, 1/n, 1/(n + 1), 1/(n + 2), 1/(n + 3), ...
>>
>> We can even refer to such unit fraction "in terms of x":
>>
>> All of the following (infinitely many) unit fractions are smaller than 
>> x: 1/ceil(1/x + 1), 1/ceil(1/x + 2),  1/ceil(1/x + 3), ...
>>
>>> Between x and any y that is different than it (x), there will be a 
>>> unit fraction to fit into the gap. infinitely many.... :^)
>>
>> Nope. There is no unit fraction (strictly) between, say, 1/2 and 1/1.
> 
> What about 1/4? Ahhhh! You mentioned the word _strictly_. Okay.
> 
> Humm... Well, if we play some "games" ;^), then 1/4 would sit in the 
> center of the gap between 1/2 and 1/1 where:

Really?

??? 1/2 < 1/4 < 1/1 ???

Are you sure?

0.5 < 0.25 < 1

Hmmm...?

>> In other words, there is no unit fraction u such that 1/2 < u < 1/1.

Concerning 1/4, in my book (of numbers):

     1/4 < 1/2 < 1/1. :-P

It's clear that you have/had 3/4 in mind. (i.e. 1/2 + 1/4. :-)

But 3/4 is't a unit fraction. :-P