The issue has been discussed many times. This proposal is primarily about the
definition of infinity.
Pythagorean's real number is Q, they could use the infinite-approaching argument
very validly deducing that all numbers are ratio number. Anyone can use Q to
approach any number and deduce that all real numbers are rational (sure modern
people won't do this).
Snippet from https://groups.google.com/g/comp.theory/c/DaybI0JY4Vc
To add more material came up to me (not well ordered):
There are quite a number of proofs of "repeating decimals are irrational".
The basic is the correct equation of 1/3 and its decimal form from long
division (kids understand this 'infinity' with no problem) should be:
1/3= 0.333... + nonzero_remainder.
To translate the 0.999... problem to limit:
Let A= lim(n->∞) 1-1/2^n = 0.999...
B= lim(n->∞) 1-1/10^n = 0.999...
<=> lim(n->∞) 1-1/2^n = lim(n->∞) 1-1/10^n
<=> lim(n->∞) 1/2^n = lim(n->∞) 1/10^n
<=> lim(n->∞) 1 = lim(n->∞) 1/5^n
[Note] I just demonstrate an instance. The limit theory can evolve as it does
(e.g. one-sided limit... There are many slightly different versions of
interpretation of limit as it evolves). Readers might find different
authors use different rules.
Limit is a technic to find its 'limit', it cannot form a logically
consistent theory for real number, e.g. the result of limit in general
must be verified, e.g. numerically, one cannot absolutely trust the
result of limit arithmetic. And at final, lim(x->c) f(c)= L does not
'deduce' f(c)=L (In text book, probably just reads "lim(x->c) f(c)= L, SO
WRITTEN as f(c)=L"). Limit theory only says the limit of 0.999... is 1,
the theory does not say 0.999...=1. There is no equality concept in the
If one resorts to Dedekind-cut-like theories (I did not really read it),
from the knowledge that all the combinations of discrete symbols cannot
represent all the real numbers, I can conclude what those theories
claim are false, let alone I suspect there should be circular arguments
there, because many terms there must be well defined as a fundamental
theory, are undefined (prove me wrong).
The limit example above demonstrated "0.999..." cannot denote a specific number,
which also means "repeating decimal" cannot specify a unique number (A!=B).
Using limit is invalid for me (for this question) but the result is correct,
see the provided reference (I found a typo there).
Simple arithmetic (this should also be a valid way 2.718... is calculated):
(0.999....)^n approaches 1/e
(1.000...1)^n approaches e (or defined as e)
A possible rebuttal might be that the (1-1/n) in lim(n->∞) (1-1/n)^n is an invalid
number (approximated like 0.999...), or it is a 'concept' etc...
But if it is not a number, the whole equation is broken.
The density property says (implicitly) n can enumerate infinitely (otherwise, it
won't be a rule) and A[∞] never be 1. A[n] infinitely approaches 1 in form
like 0.999.... This is like in the case of the interval [0,1), infinite numbers
of 0.999...s are located near the open end of [0,1).
Can we infinitely refine the scale of a ruler and the last scale never touches
the scale of 1? I think, yes, something like the √2 story, otherwise all numbers
can be 'proved' rational.