> This bothered me, too, as I have vague memories of having seen > associativity and the rest proved from the Cauchy sequence definition > in some course I attended decades ago. I don't know what to make of > this paragraph. I hope he elaborates on it sometime.
It's not difficult to show Cauchy sequences define a commutative ring, in fact an algebra. When you show null sequences are a maximal ideal, you are set. Of course if you object on principle to things like maximal ideals, you aren't set, but that is my objection to the whole enterprise--it's just a version of Mathematics Made Difficult. It's like having someone tell you you will go to hell if you don't wear the right kind of underwear; there's just nothing much attractive about making proofs harder, though finding what is the minimal assumptions you are required to make is interesting mathematics in itself.