let 0.999999.... = lim(n--> oo) (9/10 + 9/(10^2) + ..... + 9/(10^n) )

.....................= lim(n--> oo) 9/10 (1 + 1/10 + 1/(10^2) + ....)

.....................= lim(n--> oo) 9/10 * lim(n--> oo) (1 + 1/10 + 1/(10^2) + ....)

.....................= 9/10 *lim(n--> oo) (1 + 1/10 + 1/(10^2) + ....)

Note that (1 + 1/10 + 1/(10^2) + ....) is a geometric series with a = 1 and r = 1/10 (which means |r| < 1). so it's infinite sum is given by a/(1 - r)

......................= 9/10 * 1/(1 - 1/10) = 9/10 * 1/(9/10) = 9/10 * 10/9 = 1