Re: Cantor Diagonal Proof

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.

--
Alan Mackenzie (Nuremberg, Germany).