Re: Cantor Diagonal Proof

On 4/11/25 7:32 AM, wij wrote:

On Fri, 2025-04-11 at 09:50 +0000, Alan Mackenzie wrote:

wij <wyniijj5@gmail.com> wrote:

On Thu, 2025-04-10 at 17:23 -0700, Keith Thompson wrote:

wij <wyniijj5@gmail.com> writes:
[...]

"lim(x->c) f(x)=L" means the limit of f approaching c is L, not
f(c)=L 'eventually'.  f at c is not defined (handled) in limit.

>

Correct.

>

lim 0.333...=1/3    ... The *limit* is 1/3, not 0.333...=1/3
0.3+0.33+0.333+...  ... The sequence converges to 1/3
Σ(n=1,∞) 3/10^n     ... The sum converges to 1/3 (or you can use lim)

>

The limit as the number of 3s increases without bound *is exactly what
we mean* by the notation "0.333...".  Once you understand that, it's
obvious that 0.333... is exactly equal to 1/3, and that 0.333... is a
rational number.

>

You agree "f at c is not defined (handled) in limit", yet, on the other hand
ASSERTING 0.333... is 'exactly' 1/3 from limit? Are you nut?

>
No, Keith Thompson is simply correct, here.  It is you who are nuts,
making unfounded claims about mathematics without having studied it.
>
The sentence ....

The limit as the number of 3s increases without bound *is exactly what
we mean* by the notation "0.333...".

.... is entirely correct.
>

As usual, you need to prove what you say. Or you are just showing yourself
another olcott, just blink belief, nothing else.

>
No, one doesn't need continually to prove standard mathematical
definitions and results.  One could spend the whole day, every day, doing
nothing else.
>
It is _you_ who needs to prove your remarkable assertions.  You can't, of
course, because they're false.  What you could do, of course, is to show
a bit of respect for those who have studied and learnt mathematics.

 I am not interesting to blind beliefs.
As I may guess from your posts, your knowledge is essentially 'what people say'
without knowing the meaning of words.
You may say it is 'standard', 'mainstream'...,etc. But whatever it is, simply
no logical proof.
 Remind you, the so called 'standard', 'mainstream' is on the side of logical proof.
They may evolve/change from errors. It is not a static thing and not the source of fact.
 To save garbage talks, provide your logical proof (as usual, I believe NONE).
 

Remmeber, the claim is that 0.33333... is 1/3 in the limit, i.e. that for any possible epsilon, no matter how small, but still positive, there is a point in the sequence of generatation of 0.3333... that all points after that will be closer to the limit then epsilon.
We can compute that point, and thus show the limit is that value.
We do that by taking the log base 10 of epsilon, taking its floor (the largest integer that is less than or equal to the value), negate it, and use that many 3's (but at least one if we start with a big epsilon).
For instance, an epsilon of 0.001 has a log base 10 of -3, so we say that all number of the pattern with at least 3 3's are that close.
we can show the example as 1/3 - 0.333 will be 0.0003333... which is less than 0.0004 which is less than 0.001, and adding more 3s to the number just makes us closer.
Thus we have the limit, and thus the proof by the definition.
It seems your problem is you don't actually believe in the concept of limit as a rigerous mathematical process, which just means you aren't following the defintions.