If we let S(n) = .9999...9 (with n 9s)
S(n) = sigmar = 1 to n 9 * 10-r (From the definition of a base 10 number)
10S(n) = sigmar = 0 to n-1 9 * 10-r (Multiply each term of the sum by 10)
9S(n) = 10S(n) - S(n) = sigmar = 0 to n-1 9 * 10-r - sigmar = 1 to n 9 * 10-r = 9 - 9 * 10-n
S(n) = 9S(n)/9 = 1 - 10-n
limn goes to infinity S(n) = 1 - limn goes to infinity 10-n = 1 - 0 = 1
QED