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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Fri, 20 Sep 2024 18:36:18 -0700
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On 9/20/2024 6:03 PM, Moebius wrote:
> Am 21.09.2024 um 01:23 schrieb Chris M. Thomasson:
>> On 9/19/2024 2:13 PM, Moebius wrote:
>>> Am 19.09.2024 um 21:13 schrieb Chris M. Thomasson:
>>>> On 9/19/2024 5:55 AM, WM wrote:
>>>>> On 18.09.2024 22:49, Moebius wrote:
>>>>>> Am 18.09.2024 um 21:35 schrieb Chris M. Thomasson:
>>>>>>> On 9/18/2024 5:44 AM, WM wrote:
>>>>>>>> On 16.09.2024 03:16, Richard Damon wrote:
>>>>>>>>> On 9/15/24 3:39 PM, WM wrote:
>>>>>>>>>> On 15.09.2024 18:38, Ben Bacarisse wrote:
>>>>>>>>>>
>>>>>>>>>>> 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.
>>>>>>>>>>
>>>>>>>>>> That proves that small intervals cannot be defined (they are 
>>>>>>>>>> dark). 
>>>>>>>>>
>>>>>>>>> But arbitrary small intervals CAN be defined.
>>>
>>> For any eps e IR, eps > 0: [0, eps] is an interval (@WM: you see, I 
>>> just defined it) of length eps and it countains an uncountable number 
>>> of points. Hint: eps may be arbitrarily small, as long as it is > 0.
>>>
>>>>>>>>> Try to name one that can't.
>>>>>>>>
>>>>>>>> Define an interval comprising 9182024 points, starting at zero.
>>>>>>
>>>>>> There IS NO "interval comprising 9182024 points", hence NOTHING TO 
>>>>>> DEFINE HERE, you fucking asshole full of shit.
>>>>>
>>>>> How can infinitely many points be accumulated without a first one? 
>>>
>>> @WM: There's no need for them "to be accumulated" (whatever this may 
>>> mean), you fucking asshole full of shit.
>>>
>>>> The real line is infinitely long 
>>>
>>> WM is talking about some interval of finite length here, it seems.
>>>
>>>> and infinitely dense, or granular if you will...
>>>
>>> Sorta.
>>
>> Well, its infinitely long...
>>
>> ...(-1)------(0)--------(+1)...
>>
>>
>> It has no end just like there is no end to the signed integers. Also, 
>> its infinitely dense due to the nature of the reals.
>>
>> ?
> 
> Yes.
> 
> Here's a mind bending fact concerning the reals and rational numbers.
> 
> Between any two real numbers there is a rational number and between any 
> two rational number there is a real number. But there are only countably 
> many rational numbers while there are uncountably many real numbers. :-)
> 
> Math lingo: "The rationals are dense in the reals". :-)

As in there is a rational that can be generated by the infinite 
convergents of continued fractions that can be used to represent the 
target real up to an infinite desired accuracy? Fair enough?