>>16
The basic definition of convergence of a sequence over a metric space (in this case, R with d(x,y)=|x-y| is given by:
{x_n} is said to converge to x iff for every epsilon > 0, there exists an natural number N such that for all n > N, d(x, x_n) < epsilon.
Basically you take x_1 = 0.9, x_n = x_(n-1) + 9/10^n as your sequence, which clearly becomes 0.999... as n approaches infinity and show that it converges to 1.