a real number is actually an equivalence class of the limit of sequence of rational numbers
Take the sequence .9, .99, .999, .9999, .99999, ...
Obviously it converges to .9 repeating.
However, it also converges to 1.
Thus the limit of the sequence converges to both .9 repeating and 1, so by definition the two equivalence classes (and thus the real numbers) are equal.