Another proof:
0.999...
= 0.9 + 0.09 + 0.009 + ...
= 9 * 0.1 + 9 * 0.01 + 9 * 0.001 + ...
=1 -9 + (9 * 1 + 9 * 0.1 + 9 * 0.01 + 9 * 0.001 + ...)
=2 -9 + 9 / (1 - 0.1)
= -9 + 9 / 0.9
= -9 + 10
= 1
(1): I subtracted 9, and added 9 in in the form of 9 * 1, plus I added parentheses. Makes no difference because -9 + 9 = 0, which makes absolutely no difference in the sum.
(2): It's a geometric series, whose sum is well known. In case you didn't know, a geometric series is a series that has the form a + ar + ar^2 + ar^3 + ..., and its sum is known to be a / (1 - r), assuming|r| < 1. The sum formula is quite easy to prove and there's nothing special about it, so there shouldn't be anything controversial going on there. Here we simply have a = 9 and r = 0.1.
Anyway, the whole thing is because of the decimal notation. Nothing special about the numbers, it's just the notation that fails at this level.
