Asymptotic analysis is only relevant if you have some series (or a function).
The series 0.9, 0.99, 0.999,... is asymptotically close to 1 and also asymptotically close to 0.9999... (with infinite 9s), since for any epsilon, I can find an index N after which all elements of the series are within epsilon of the target.
Since a single series can't have two limits, 1.0 should be equal to 0.999...
Note that real numbers are allowed to have infinite digits after the point, otherwise they wouldn't include things like 1/3.
The series 0.9, 0.99, 0.999,... is asymptotically close to 1 and also asymptotically close to 0.9999... (with infinite 9s), since for any epsilon, I can find an index N after which all elements of the series are within epsilon of the target.
Since a single series can't have two limits, 1.0 should be equal to 0.999...
Note that real numbers are allowed to have infinite digits after the point, otherwise they wouldn't include things like 1/3.