Jorfer s complaint above is actually deeper than just whether or not 0 999 1 His issue rather is rather the concept of limits actually applies to real life He is agreeing that the series 0 9 0 99 0 999 has a limit of 1 but he is not agreeing that the corresponding decimal 0 999 should be said to be equal to 1 but that they must instead differ by some as yet undefined infinitesimal amount The reason that an infinite series in the real number system has a sum equal to the limit of its corresponding partial finite sums is that the real number system is defined in such a way that two numbers which differ by at most an infinitesimal quantity must be equal The article Archimedean property has a section that explains why the reals have no non zero strictly positive infinitesimal numbers Since 0 999 and 1 differ by at most an infinitesimal quantity and the only such quanity in the reals is zero 1 0 999 0 and thus 1 0 999