This is very good. It involves a level of topology beyond my casual competence.
I suppose you are either blissfully kind and youthful, or outright mocking me, and will assume the former because you don't seem like the mocking kind.
But really -- my post, in itself, does not. Where you're proposing to go, however... More in a minute.
0.999... is in every neighborhood of 1.
Let's accept that definition: Take the metric space (X, d) with X = R and d being any metric. You are saying that for all epsilon > 0 it holds that d(0.999..., 1) < epsilon. This implies that d(0.999..., 1) = 0, thus 1 = 0.999... is proven by definition of a metric, which includes the criterion d(a, b) = 0 <=> a = b. Your only recourse is to introduce the "infinitesimal" number (defined as greater than zero, but smaller than any positive real number, and necessarily not real in itself since you could just plug it in as the epsilon otherwise, apart from the problems I've already mentioned). Then the fun begins -- you must construct your own version of analysis on an algebraic structure that must differ from R in
at least one major property:
a) the Archimedean axiom does not hold; or
b) it is not Cauchy-complete; or
c) it is not ordered; or
d) it is not a field.
Amusingly enough, the construction that you're heading towards by axing the first property (hyper-real numbers), whether you know this or not, incorporates another "close enough" promoted to a relation of equivalence, so as to ensure that the resulting structure is a field (which necessitates ab = 0 <=> a == 0 or b == 0), by identifying the sequences (a_n) and (b_n) with each other iff |{n in N | a_n != b_n}| < infinity. Which is precisely what you're arguing against. The idea is to extend R into a field of sequences by defining r := (r, r, r, ...) for any real number r, and then identify the sequences that converge to infinity with "hyper-reals" which grow larger than any real number.
(Pure semantics: even then, 0.999... == 1 stays defined as the limit of the convergent series etc. as usual, but the sequence (0.9, 0.99, 0.999, ...) is identified with precisely what you're proposing.)
(If we accept the infinitesimal number, henceforth called "i" -- not like that letter is reserved for anything important -- as real, then against all intuition, the sequence (1, 1/2, 1/3, 1/4, ...) = (1/n) would not converge on the usual metric space (R, d), where d(a, b) = |a - b|. Nor, in fact, could any sequence converge except those with a constant subsequence. -- Proof: Let i be defined such that 0 < i < r for any positive real number r. Let (a_n) be a sequence for which there exists no n_0 such that for all n > n_0: a_n = c. Let epsilon := i. Then for any n_0 and any prospective limit x, there exists an n > n_0 for which d(a_n, x) = |a_n - x| is real, thus > i > 0. -- This is one of the consequences of infinitesimal numbers: metric spaces no longer suffice for analysis; you need to know topology to survive. Other "common-sensical" notions, such as the existence of a supremum/infimum on all sets bounded from above/below, which the real numbers provide for, do not apply to the hyperreals either.)
In a nutshell, it is
possible to define such a system, even fruitful -- but you must acknowledge that you're abandoning the analysis of real numbers in favour of something decidedly more counter-intuitive and absurdly inaccessible to anyone but mathematicians, unlike real analysis. Within the realm of real analysis, 0.999... == 1 directly follows from the definition of a series as the limit of its partial sums, as has been stated.
Unfortunately, most people who aggravate themselves against mathematics in school will never even hear the term "series". This is the true problem. I presume that for many such students, the recourse is not to head towards non-standard analysis, but rather to shut themselves off from any effort to understand mathematics, in frustration at something perceived as arbritrary, against common sense, "paradoxical" etc.
