1.5. THE CONSTRUCTION OF THE REAL NUMBERS 11 Exercise 1.5.9. Show that, with addition and multiplication defined as above, C is a commutative ring with 1 (see Appendix A). Now let I be the subset of C consisting of sequences (ak)k∈N with the property that, given any rational r 0, there exists an integer N such that if n ≥ N, then |an| r. The set I consists of Cauchy sequences which converge to 0. Suppose (ak)k∈N / I. Then there exists an r 0 such that |ak| ≥ r infinitely often. Pick N ∈ N such that |an − am| r/2 for n, m ≥ N. This implies that |an| |am| − 1 2 r for n, m ≥ N. Fix an m ≥ N for which |am| ≥ r. Then for all n ≥ N, we have |an| 1 2 r. Thus, Cauchy sequences which do not converge to 0 are eventually bounded below (in absolute value) by some positive constant. Exercise 1.5.10. Show that if a Cauchy sequence does not converge to 0, all the terms of the sequence eventually have the same sign. Definition 1.5.11. Let (ak)k∈N and (bk)k∈N be Cauchy sequences in Q. We say that (ak)k∈N is equivalent to (bk)k∈N, denoted by (ak)k∈N ∼ (bk)k∈N, if (ck)k∈N = (ak − bk)k∈N is in I. Exercise 1.5.12. Show that ∼ defines an equivalence relation on C. Exercise 1.5.13. Given a ∈ Q, show that the collection of Cauchy sequences in C converging to a is an equivalence class. In particular, I is an equivalence class. Denote by R the set of equivalence classes in C. We claim that, with ap- propriate definitions of addition and multiplication (already indicated above) and order (to be defined below), R is an ordered field satisfying the least upper bound property. If (ak)k∈N is a Cauchy sequence, denote its equivalence class by [ak]. As one might expect, the sum and product of equivalence classes are defined as follows: [ak] + [bk] = [ak + bk] and [ak][bk] = [akbk]. Exercise 1.5.14. Show that addition and multiplication are well-defined on R. Exercise 1.5.15. Show that R is a commutative ring with 1, with I as the additive identity and [ak] such that ak = 1 for all k as the multiplicative identity. This follows easily from Exercise 1.5.9.

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