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.999999... is equal to 1, exactly. Here is the simple proof:
1/9=.111111... (easy to prove using Geometric Series)
.999999...=(9)(1/9)=1
This fact (.999999...=1, along with similar equations like .1249999...=.125) slightly complicates a few proofs in set theory. When dealing with all possible decimal representations, you may have to show that you are accounting for the ones that are redundant. We deal with the redundancy by outlawing any decimal which ends in infinitely many nines (replacing them with their equivalent non-repeating decimal).
I received email suggesting that the above proof may be easier with 1/3=.333333... as the first step. It may be more obvious, as more people know that 1/3=.333333... But the proof is virtually identical then because, even though we all know that 1/3=.333333..., we still need to use a geometric series to prove that. In my opinion, 1/9=.111111... is slightly more esthetically pleasing, for some reason.