Sure we do. 1/3 is represented in decimal as zero followed by a decimal point followed by an infinite number of 3's. That's a perfectly valid representation, just like 1/2 is zero followed by a decimal point followed by 5 followed by an infinite number of 0's. Similarly, the number 0.999... is zero followed by a decimal point followed by an infinite number of 9's. Any real number that you can add to this that would be greater than 1, so 0.999... = 1.

EDIT: And this representation is really just an artifact of using base 10. For example, in base 9, 1/3 is exactly 0.3. The fact that it doesn't have a finite representation in decimal doesn't mean it's an approximation.