>>11490813>>11499827>meaning "..." merely defines an arbitrary but finite amount of repetition...which is, exactly as you say, *defined* as (formally) whether for all ? > 0, we can choose an integer N such that for any integer n >= N, the nth term in the sequence (here, 0.999...9 with n '9's) differs by less than ? from the value we're assigning it (that is, 1). This is dealing only with finite numbers, because (at least in the system of real numbers we're working with here) those are the only kind that exist.
>and so 1 - 0.999... = 0.000...1 is a true statement>no more or less than 1 - 0.999 = 0.001And so we come to our conclusion. Again, you're close. 0.000...1 exists for a finite number of zeros. But I think you'd agree it's pretty trivial to find, for some ? > 0, some n such that 10^-n < ?; that is, for any ?, eventually the sequence 0.1, 0.01, 0.001, 0.0001... will have a smaller number, and all subsequent numbers in the sequence can be trivially shown to be smaller than this number. And from this, the definition of a limit puts the sequence 0.1, 0.01, 0.001, ... as converging to 0, and 0.9, 0.99, 0.999... as converging to 1.
>and the assumption that 0.999... = 1 is proven false.A quick note: if 1 - 0.999... = 0.000...1, and if our hypotheses are correct, then it's equivalent to 1 - 1 = 0. Your point here is, if anything, a point of guidance in our favor.
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The above's an analytical, by-the-book answer: this is the definition of a limit, and infinite decimal representations are defined in terms of limits, so that's that.
I would mostly like to ask you: if you disagree with this analysis, then not only where, but also why? What does the concept of 0.000...1 mean, if not zero? If it's a specific number that's not zero, then you could find an even smaller number by halving it; that's not "infinitely" anything to me. Just because the sequence 0.1,0.01,0.001,... is all positive doesn't mean that the value it converges to is positive.
