Infinity is not a number. .999... is. Therefore, this is irrelevant. I'm not sure how sqrt(2) relates, but I'll let it pass.

You're considering finite expansions. Finite amounts are irrelevant, because .999... is an infinite expansion. The fact that the distance(amount you must add) between them is an infinitesimal means they are equal. Since, apparently, brewbuck's statement isn't enough, here's a link:

http://www.calvin.edu/~rpruim/courses/m361/F03/overheads/real-axioms-print-pp4.pdf

Name ONE rational between .999... and 1. That's all I want. Anything else is intuition, and math is not based on intuition.Theorem 0.22: [The rationals are dense in the reals.]

Between any two distinct real numbers there is a rational number. (In fact,

there are infinitely many such rational numbers.)

"Axioms" are true by definition. I would use "proof" or "statement" in this case, IMHO.

Yeah, oops, we've reached the end of an infinite expansion and shifted it. Big oops.

"indeterminate" is the word you're looking for, and 1-.999... isn't indeterminate. Again, .999... is a number, not infinity.

It never reaches anything, it's a constant. 1 doesn't reach 1, it is 1. Refer to the quote of the theorem above for a rigorous proof.