0.9(9) =     Sum         9/10^n 

                          (n=1 -> Infinity)

                    =  lim               sum      9/10^n

                     (m -> Infinity) (n=1 -> m)

                    =  lim           .9(1-10^-(m+1))/(1-1/10)

                     (m -> Infinity) 

                    =  lim           .9(1-10^-(m+1))/(9/10)

                     (m -> Infinity) 

                    = .9/(9/10)         

                    = 1