You can also prove it a different way if you allow the use of the formula for finding the limit if 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