I should probably start my answer with a question myself. What do you mean by "repeat the nines to infinity"? I understand the number 0.99999... where there is a well-defined finite number of nines, but infinity is NOT a number. Therefore, the expression 0.99999 has no meaning unless we give it one. To define what the expression 0.999...inf (or equivalently 0.999r) means, we define the sequence a_n as follows:
a_1 = 0.9, a_2 = 0.99, a_3 = 0.999, ... or equivalently a_n = 0.999 ... n nines ... 999
and we DEFINE 0.999r as the LIMIT of this sequence, where the limit of a sequence (assuming it has a limit) is a number L such that we can get a_n arbitrarily close to L simply by choosing a sufficiently large value for n. In the case of 0.999r, we see that no member of the sequence a_n equals 1, but we can get a number as close to 1 as we like simply by writing enough nines in the number 0.99999... Therefore, we can say that the limit of the a_n sequence is EQUAL to 1, or equivalently 0.999r = 1.