0.9999999999999999999999999 is like saying
∑(9/10^n,n,1,k)
so take the limit of k 
∞ and you get 1
therefore 0.999999999.... = 1http://img408.imageshack.us/img408/6194/msp428419b3e84ghc5791ge.gif
an equivalent equation would be
lim_(k->infinity) (10^k-1)/10^k
Simplify (10^k-1)/10^k assuming k>0 [which it is, starts at 1
∞] giving 1-1/10^k:
= lim_(k->infinity) (1-1/10^k)
The limit of a difference is the difference of the limits:
= 1-lim_(k->infinity) 1/10^k
The limit of a quotient is the quotient of the limits:
= 1-1/(lim_(k->infinity) 10^k)
Using the continuity of 10^k at k = infinity write lim_(k->infinity) 10^k as 10^(lim_(k->infinity) k):
= 1-1/10^(lim_(k->infinity) k)
The limit of k as k approaches infinity is infinity:
= 1
you take the limit of that and you get .999999...