Hmm, this doesn't have to be looked at through vector calculus.

0.999... = 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + 9(1/10)^4 + ...

That sum is an infinite geometric series, and since the absolute value of it's common ratio is less than 1, it converges. So, you can calculate it's sum. It's involves a lot of Reimann sum symbols and other things that would involve annoying use of the character map, but it boils down to this:

ar + ar^2 + ar^3 + ... = ar/(1-r)

9(1/10) + 9(1/10)^2 + 9(1/10)^3 + ... = 9(1/10)/(1-(1/10)) = 1

So, 0.999... = 1. People can argue someone proving that with limits and say something about the nature of limits to infinity and their thoughts on infinitesimals, but the convergence don't lie.