[Sharlin]

It can, and infinitesimals are a part of so-called nonstandard analysis, but you cannot write an infinitesimal using decimal notation. "0.999…1" is simply meaningless, a contradiction. If you have a "…" it means there's no place where you could put a "last" digit. If "0.999…1" doesn't feel impossible enough, then what would "0.999…9" mean?

[hinkley]

People in this thread seem to think that “9 repeating” is not an infinity of nines but is instead “write or think of nines until you get bored and then write something else”