Sigh. Here we go, a definition of the real numbers. I'm only doing this once. Let's look at the set of all cauchy sequences of rational numbers. A sequence

Sigh. Here we go, a definition of the real numbers. I'm only doing this once. Let's look at the set of all cauchy sequences of rational numbers. A sequence of rational numbers is just a map N->Q. We write a_0, a_1, a_2, ... for the sequence which maps 0|->a_0, 1|->a_1, 2|->a_2, ... Such a sequence is called "cauchy" if it has the property that: For all epsilon > 0 there exists an N such that n,m > N => |a_n-a_m|<epsilon. Basically, this means that the tail of the sequence bunches together. Now, we define an equivalence relation ~ on the space of cauchy sequences. We say: a_0, a_1, a_2... ~ b_0, b_1, b_2... if: a_0-b_0, a_1-b_1 ,a_2-b_2... converges to zero (A sequence a_0,a_1,... converges to zero if, for every epsilon > 0, there exists N such that n>N => a_n < epsilon.) Now, the real numbers are the set of equivalence classes of rational cauchy sequences under ~, as defined above. Clearly, .999... is (the equivalence class of): 0.9, 0.99, 0.999, ... And 1 is the equivalence class of: 1, 1, 1, ... So are these the same? Well, we have to check if our representatives are equivalent. Does this sequence converge to 0? 1-0.9, 1-0.99, 1-0.999, ... Yes, it does. So 1 = .999...