This does not occur over time. There is no "approaching" here. If we were talking of a series such as .9, .99, .999, .99999999999999999999, .99999999999999999999999999999999999999999999, you would be correct, but .999... represents those 9s extending into infinity, in the present, right now, forever.
Thus equaling 1, by your own admission.
.999_ is less than 1 by an infinitely small difference, but a difference nonetheless.