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For any number that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than . So, it does not make sense to identify 0.999... with any number smaller than 1.
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The correct notation is (0.9; 0.99; 0,999; ...). Q: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly?
Near the end "This implies that the difference between 1 and x is less than the inverse of any positive integer." This is not true. That symbol is less than or equal, which means that the difference between 1 and x can be equal to the inverse of any positive integer, which includes options that are not 1, so the proof is not complete.
OK... I learned about 0.(9) being the same as 1 in high school too... But now I have this question. 1 = 0.99999... right? 2 = 1.99999... every ok so far. does this ...
(For instance, it's less than 0.0001 because 0.999999... > 0.9999 and 1 − 0.9999 = 0.0001, and you can make this argument by truncating the expansion 0.999... after any number n of digits and see that it's less than 10 −n.) It also obviously can't be negative because 1 can't be less than 0.999... (I trust this bit is obvious).
If this conception of induction is valid, then we could argue as follows: Let p_i be the assertion "0.9999...999, with i nines, is less than 0.999..." For i = 1 we have 0.9 < 0.999..., which is clearly true. Now suppose p_k is true. Now, clearly, 0.9999...999, with k + 1 nines, is less than 0.999... (This doesn't require the hypothesis that p_k ...
For Dedekind cut's, I'd say that any rational number less than 1 is also less than .999..., and (obviously) vice versa.) Mango juice talk 20:48, 29 September 2006 (UTC) I would love to be able to write about the importance of 0.999… and its history. Unfortunately, it isn't important, and it has no history. I'm not kidding.