You can also prove it a different way if you allow the use of the formula for finding the limit of the sum of a geometric series on a non-convergent series.

Sum(ar^n, n=0, inf) = a/(1-r)

So,

…999999

= 9 + 90 + 900 + 9000…

= 9x10^0 + 9x10^1 + 9x10^2 + 9x10^3…

= Sum(9x10^n, n=0, inf)

= 9/(1-10)

= -1