Re: How many different unit fractions are lessorequal than all unit fractions?

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.

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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.

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Define an interval comprising 9182024 points, starting at zero.

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There IS NO "interval comprising 9182024 points", hence NOTHING TO DEFINE HERE, you fucking asshole full of shit.

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How can infinitely many points be accumulated without a first one?

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@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

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WM is talking about some interval of finite length here, it seems.
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and infinitely dense, or granular if you will...

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Sorta.

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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?