Re: Definition of real number ℝ --infinitesimal--

Op 30.mrt.2024 om 14:56 schreef olcott:

On 3/30/2024 7:10 AM, Fred. Zwarts wrote:

Op 30.mrt.2024 om 02:31 schreef olcott:

On 3/29/2024 8:21 PM, Keith Thompson wrote:

olcott <polcott2@gmail.com> writes:

On 3/29/2024 7:25 PM, Keith Thompson wrote:

[...]

What he either doesn't understand, or pretends not to understand, is
that the notation "0.999..." does not refer either to any element of
that sequence or to the entire sequence.  It refers to the *limit* of
the sequence.  The limit of the sequence happens not to be an element of
the sequence, and it's exactly equal to 1.0.
>

In other words when one gets to the end of a never ending sequence
(a contradiction) thenn (then and only then) they reach 1.0.

>
No.
>
You either don't understand, or are pretending not to understand, what
the limit of sequence is.  I'm not offering to explain it to you.
>

>
I know (or at least knew) what limits are from my college calculus 40
years ago. If anyone or anything in any way says that 0.999... equals
1.0 then they <are> saying what happens at the end of a never ending
sequence and this is a contradiction.
>

>
It is clear that olcott does not understand limits, because he is changing the meaning of the words and the symbols. Limits are not talking about what happens at the end of a sequence. It seems it has to be spelled out for him, otherwise he will not understand.
>

 0.999... Limits basically pretend that we reach the end of this infinite sequence even though that it impossible, and says after we reach this
impossible end the value would be 1.0.

No, if olcott had paid attention to the text below, or the article I referenced:

https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

he would have noted that limits do not pretend to reach the end. They only tell us that we don't need to go further than needed and that this is reachable for any given rational ε > 0. It is interesting that this is sufficient to construct reals.

 

0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 = 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The = symbol in the context of a limit means in this case:
For each rational ε > 0 (no matter how small) we can find a number N {in this case 10log(1/ε)}, such that for all n > N the absolute value of the difference between xn and 1.0 is less than ε.
It is not more and not less. Note that it does not speak of what happens at the end of the sequence, or about completing the sequence.
If olcott wants to prove that 0.999... ≠ 1.0 (in the real number system), then he has to specify a rational ε for which no such N can be found. If he cannot do that, then he is not speaking about real numbers.
>

Good to see that there is no objection against this proof.