In mathematics, a definition cannot be 'wrong'; it can lead to an inconsistency, but that isn't the same thing.
The definition of the real numbers (the set of numbers where 0.999... == 1) does not lead to any inconsistency. However, there are other consistent definitions of sets of numbers where (the equivalent of) 0.999... does not equal (the equivalent of) 1. Those sets of numbers have values mathematicians call 'infinitesimals', which do not exist in the set of the real numbers. The hyperreals are one such set of numbers.
