Indeed. The sequence is convergent. It is said the sequence approaches L. Or L is the limit of the sequence. Or more appropriately, the sequence converges to L.Originally Posted by whiteflags
Nowhere, though, does this realization establish the true identity of the sequence. Merely, its L.
For convenience we can safely use L in our algorithms (certainly for almost any use of Reals). But as a means to prove the true identity of a sequence, limits do nothing for you. L is not the indentity of a sequence anywhere else but within the domain of limits.
As a means to prove the identity of 0.999... this is just not good enough. You don't prove that there's a town hidden behind a mountain by increasing the magnification of your binoculars.
EDIT: In fact, if we consider the definition of 0.999... as that of a number that converges to 1 at infinity, it becomes evident that 0.999... is a number that will never be 1. Limits propose to give this number an identity still, for convenience sake. And that's fine. But they are not rigorous proofs of a the identity of a sequence. And this is why one must look at the context of the problem being proposed, before hastily mixing the equality signs of a limit and that of algebra. For this particular purpose of proving that 0.999... equals 1, that mixing is unacceptable.
EDIT2: And before you say "but it is for Reals!", no it is not! Exactly because we had problems with this limitation of Reals, extensions were proposed to include the notion of an infinitesimal into R. There's two solutions: A new number system (Hyperreals) which we ignore since it's not R, although it behaves just like R in ZFC. But alternatively, an extension to the language and the axioms (Internal Set Theory) which introduces new axioms and language in R. You can call it R 2.0. A new and revised R. A better and more complete R. One that stops this nonsense of trying to prove that a number that can never be 1 is 1.
