First we must be recognize that a real can be written in any number of ways. The notations are obviously not equal, but it may be that the represented numbers are the same.
The next step is to get clear about the meaning of 0.999..., and that will ultimately require some understanding of the construction of the reals. Put a(i)= 9/10^i, then
0.999... = lim n->inf (a(1)+ a(2)+ ... + a(n))
= lim n ->inf (1-1/10^n)
by definition. This limit can be shown to exist. Moreover the limit is clearly greater than any real less than 1 and clearly less than or equal to 1. The conclusion? The limit must be equal to 1.