RE: I will bite, a defense of sorts

Actually, 1 = 0.999.... It is not an approximate statement. Hard to grasp, yes, but no approximations have been made. It is easier to look at 0.111... first, it is simpler to deal with. To get, 0.999..., you just have to multiply the result by 9.

Now, 0.111... is really the series 0.1 (1 + 0.11 + 0.12 + ...) which is a geometric series. In generic terms, a geometric series is written a + ar + ar2 + ..., and in our case a = r = 0.1. For a finite number of terms, n, it is relatively easy to show that the sum of the series is simply

a (1 - rn + 1) / (1 - r).

For an n number of terms of the series 0.111..., the sum would reduce to

(10n + 1 - 1)/(9 10n + 1),

with the result of 11/100, 111/1000, 1111/10000, ... for n=1,2,3, ...

When n is allowed to grow large, i.e. approaches infinity, this sum will converge iff r < 1. To determine what it converges to, break the fraction in the generic formula into two pieces: one with a as its numerator, and the second with a*rn + 1 as its numerator. Only the second piece depends on n at all, and it approaches 0 as n approaches infinity. Since, the first piece does not depend on n, it will remain constant as n approaches infinity. Hence, the infinit series will converge to

a / (1 - r).

For a = r = 0.1, the resulting sum is just 1/9, which we have just shown is equivalent to 0.111.... From this you can see directly that 1 = 0.999... = 1.000..., implying that decimal expansions are not unique. While they still may not be comfortable ideas, the concepts of limits and convergence are well founded. It just took until Newton and Leibniz to make the leap necessary, solidifying the ideas.

I encourage you to look over the proof for sum of the finite series, as it is a very clever technique that should be in any mathematician's tool box.

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