True, you cannot use infinite values in algebra, but we aren't. 0.999... isn't infinite, it is infinitely long. There is a big difference. Technically, all values can be represented as infinitely long:
1 = 1.000...
3.64 = 3.64000...
Etc.
There are various proofs that 0.999... = 1, not all of them are algebraic.
1 = 1.000...
3.64 = 3.64000...
Etc.
There are various proofs that 0.999... = 1, not all of them are algebraic.