On the other hand, for any finite value of N, the sum [1] is equal to 0.9 * (1-(0.1)^N) / (1-0.1) = 1 - (0.1)^N. This value goes to 1 as N goes to ∞.
More rigorously, for any positive value, ε, there is a value N, such that the value of the finite sum is within ε of 1.
For one thing, 1 - 0.999... = 0.000... because you never get to have any remainder since 0.999... is infinite.
Or here's another proof:
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9.000... = 9
9x = 9
x = 1
You can't represent 0.2 exactly using IEEE floats, either, but that doesn't mean the representation 0.2 is not exactly equal to 1/5th.