"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...  

"