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

On Sat, 2024-03-30 at 11:34 -0400, Richard Damon wrote:

On 3/30/24 10:57 AM, wij wrote:

On Sat, 2024-03-30 at 10:01 -0400, Richard Damon wrote:

On 3/30/24 9:56 AM, olcott wrote:

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.

 
Nope. Shows you don't really understand what limits are.
 
And are just a pathological liar as you insist that you falsehoods based
on the wrong definitions are the truth.
 

 
You are nut who always think he is talking B while reading A. (x!=c)
https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
 
 
Limit is defined on existing numbers, it cannot define the the number it is using.
Things is very simple: "repeating decimal" means the pattern is infinite.
(you are worse than olcott in this)
If it does not exit, your math (repeating decimal is ...) is garbage talking
about something does not exist and use it as proof of fact.

 
 
What number does the representation 0.abc represent?
 
it is BY DEFINITION 0 + a * 10^-1 + b * 10^-2 + c * 10^-3
 
what number does the representation 0.aaa... represent:
 
The value of lim(n-> inf) Sum(9 * 10^-i) [for i = 1 to n]
 
If a = 9, what number is that 0.999.... but also the number 1.0 since
they are the same.
 

I should have provided all that can explain your doubt, but you still
keep insisting 0.999...=1 with no proof, like POOP.

For ANY e > 0, there exists an N that for all values of function/series
witn n >= N the difference between the function and 1 is less then e.
 
BY THE DEFINITION OF LIMIT, that means that 0.999... IS EQUAL TO 1.000
 

See the link above. limit says the limit of 0.999... is 1, not 0.999... is 1.
You keep talking RD's POOP.

For Reals
 

For your POO Real (not even the obsolete real)

Remember n-ary representations are NOT numbers, but representations of
the number.
 
the value of repeating n-ary representations are defined by limits
 

I already said, limit is defined on existing number system, it cannot define
numbers it talks about.

Limits are NOT on "a number" but on a function or series (which is a
sort of function of the number of terms being used).
 

Ok, now you changed to another excuse. So, you are not really talking
about numbers, right.

 

 

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.