Am Montag, 10. Februar 2014 15:32:56 UTC+1 schrieb thenewc...@gmail.com:
> For example: lim (x->oo) (1-1/(10^x)) = 1 does not mean the expression (1-1/(10^x)) ever attains the value of 1. It NEVER does!
> It means that no matter how large x becomes, the value of the expression can never attain the value of 1.
I see you do not like to call expressions with infinite sequences of non-zero-digits numbers. That is correct, in principle, because without actual infinity, which is a self-contradictory concept, there is always a difference between 0.999... and 1. The idea that "in the infinite" the complete sequence of nines could be used as a substitution for 1.000... has deplorably lead to Cantor's mistake.
Nevertheless we can understand "0.999..." as a shorthand for "the supremum of the increasing sequence 0.9 0.99 0.999 ..." i.e., for the smallest number that will never be surpassed, not even be reached. And that is 1 with no doubt. To call it a number is a matter of taste or convenience. You can simply use it as a definition. Then nothing need be proved.