Here's one way of presenting this formally. Define a 'decimal' to be an ordered sequence of integers (called 'digits') a_1, a_2, a_3, and so on. (By 'and so on', I formally mean that for each positive integer k we have a digit a_k at the kth position in the sequence). Let's say each a_k has to be between 0 and 9 inclusive.
For each positive integer k, define the 'kth partial sum' of the sequence to be the sum from j=1 to j=k of (1/10^j) x a_j.
I'll skip over what it means if we say that the partial sums converge as k->Infinity, because it sounds like you understand what limits are and how they work. If not, I'd be happy to explain.
Now, if the partial sums converge to some value 'd', we say that the decimal has value equal to d. It can be shown that any decimal has at most one such d (which is good, because a decimal shouldn't have two values).
Now, I think you'd be happy to say that when we write 0.9999999..., what we mean is the decimal where a_k=9 for all k. Given this definition, it follows that the value of the decimal is precisely 1, using properties of geometric sequences.
It is up to you how you define '...', but all mathematicians would agree that 0.99999... should be interpreted as above if it is to have any meaning at all.
If you really want a more precise eplanation of '...' at the end of a truncated decimal, I would provide the following: "Writing 0.abcdef... asserts that the digits abcdef of the truncation provided have a pattern which should be obvious to the reader. Assign the first few digits a_1, a_2 etc as per the part of the decimal that is explicitly given; and then assign all subsequent digits values according to said pattern." - it's not a formal notation, as I say, but rather a convenient shorthand that is understood by working mathematicians. It is always possible to be more precise if one has to be.
Mathematical definitions aren't like dictionary definitions. A closed ellipsis:
1...100
An open-ended ellipsis is accepted as meaning an infinite repetition of what precedes it. So this ellipsis:
0.9999...
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.
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.
I am one of the people who don't accept the above as truth. Whether it is truth or not depends on the formal meaning of the ellipsis ... which no one provides to me (I have asked several people about this). Without a formal definition, the statement is meaningless to me.
Here is a way to present it formally.
0.9999... is sum of a geometric series a + a * d + a * d^2 ... a * d^n ... with a = 0.9 and d = 0.1.
In the limit n -> infinity, the above sum is 1. This is however only true in the limit.
It is not clear/obvious to me that ... really means "in the limit".