I should hope you're capable of working out 1/9 on paper and seeing for yourself, BoB You're essentially asking to prove 1/7 is 0.142857142857... with the 142857 pattern repeated infinitely, which is the same principle really. The repeated patterns in 1/9 and 1/7 (and other recurring numbers) occur because of a repeating sequence of remainders to carry over to the next place value when you do the division on each successive place value. Since the remainders repeat, obviously the digits in the answer will have to repeat in exactly the same way once all the place values after a certain point are the same (which is all place values after the decimal point in integers, which will all be 0), and as you've now got a repeating sequence, it's sufficient to show it occurs infinitely

[q]I believe, using the ... strategy instead of the other dot thing, the number between .999... and 1 is 0.0...01[/q]a) I think you mean 0.9...01
b) 0.9...01 is invalid anyway, since the ... means forever, and you can't stick something on the end after forever, otherwise you imply it has an end, which is contradicting the ... in the first place