Other than pointing to the Wikipedia article (thanks for the same), you have not said anything that I did not. The repetition of the decimals before the ellipsis is the same as the geometric series representation I mentioned. The Wikipedia article also mentions the limit in one of the proofs (though I am yet to read the other proofs there, which may be illuminating! :-)
Mathematics isn't politics or law. The point is to come to a consensus on the meaning of symbols and conventions. The consensus is that the ellipsis in 0.999... extends out to infinity.
In mathematics, there is every incentive to agree on axioms like the meaning of the ellipsis. In science, there is every incentive to agree on the meaning of evidence and how it relates to theory. Only in psychology do they vote on the meaning of their "scientific" findings (all the new conditions that made their way into the new DSM got there by votes, not evidence) -- but psychology isn't a science, it's a pseudoscience.
Meanwhile, based on the agreed-upon axioms, 0.999... is equal to 1.
Mathematical definitions aren't like dictionary definitions. A closed ellipsis:
Is accepted as meaning a list of all the intermediate values.An open-ended ellipsis is accepted as meaning an infinite repetition of what precedes it. So this ellipsis:
Means an infinite decimal sequence of 9's. And .999... really is equal to 1.http://en.wikipedia.org/wiki/0.999...
A quote: "In mathematics, the repeating decimal 0.999... (sometimes written with more or fewer 9s before the final ellipsis, or as 0.9, , 0.(9)) denotes a real number that can be shown to be the number one. In other words, the symbols "0.999..." and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience."
Remember that mathematics, like science, avoids ambiguity where possible. And mathematical notation strives for less ambiguity than mathematics itself.