Why .999... = 1, A Proof

Recall for any convergent geometric series with common ratio r, we have,

SUM( ar^k, k = j, j + 1, j + 2, ...) = ar^j/(1-r).

.999... =
.9 + .09 + .009 + ... =
9/10 + 9/100 + 9/1000 + ... =
9/10 + 9/10^2 + 9/10^3 + ... =
SUM( 9*(1/10)^k, k = 1, 2, ...) =
9*(1/10)/(1-1/10) =
(9/10) / (9/10) = 1

Done!