Re: Cantor Diagonal Proof

On 11/04/2025 13:11, wij wrote:

On Fri, 2025-04-11 at 12:04 +0000, Mr Flibble wrote:

On Fri, 11 Apr 2025 19:52:20 +0800, wij wrote:
>

On Fri, 2025-04-11 at 04:21 -0700, Keith Thompson wrote:

wij <wyniijj5@gmail.com> writes:

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

>
Keep the insults to yourself.  Last warning.

>
I still think 'nut' is a common word, at least a terse word for people
saying one thing and doing the other (or a liar more appropriate?)
>

My assertion is simply about what the "..." notation means.
>
Do you agree that the limit of 0.3, 0.33, 0.333, as the number of 3s
increases without bound, is exactly 1/3?  (You said so above.)

>
Increases without bound -> yes is exactly 1/3 -> no such logic
>

What exactly do you think the notation "0.333..." means?  I and many
others use that notation to mean the limit, which you agree is exactly
1/3.

>
Is this a lie? I have always consistently claiming "repeating decimals
are irrational".

>
The decimals only repeat in certain bases:

 Agree

So is it your claim that 1/3 is irrational when represented in base 2, rational in base 3, irrational in base 4...? Seriously?
Numbers are what they are regardless of how they are expressed. 1/3 is a ratio of two integers and is therefore rational by definition. The decimal expansion of 1/3 is 0.3r. Therefore, 0.3r is rational.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
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