How can you subtract 0.9999... from 9.9999 digit by digit if the digits have no end? You will never solve the equation. The numbers have no end, therefor the equation has no end.
You don't have to subtract it digit by digit.
(1) 9.999... -9 = 0.999...
(1) can be accepted because we only have to subtract one digit, the integer nine. So there aren't an infinite number of digits to subtract.
(2) But we also know that it's true that if a + b = c, then a -c = -b.
Combine these (1) and (2) and you get:
(3) 9.999... -0.999... = 9
I don't have to perform the subtraction digit per digit in infinity to know this is true, because I know it follows logically from (1) and (2), both of which don't include subtracting of digits in infinity.