10 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS Also recall that the absolute value on Q satisfies the following three proper- ties. (1) For any a ∈ Q, |a| ≥ 0, and |a| = 0 if and only if a = 0. (2) For any a, b ∈ Q, |ab| = |a||b|. (3) For any a, b ∈ Q, |a + b| ≤ |a| + |b| (triangle inequality). Exercise 1.5.1. Show that, for any a, b ∈ Q, we have ||a|−|b|| ≤ |a−b|. Definition 1.5.2. A sequence (ak)k∈N of rational numbers is a Cauchy sequence in Q if, given any rational number r 0, there exists an integer N such that if n, m ≥ N, then |an − am| r. Definition 1.5.3. A sequence (ak)k∈N converges in Q to a ∈ Q if, given any rational number r 0, there exists an integer N such that if n ≥ N, then |an − a| r. Sometimes, we just say that the sequence (ak)k∈N converges in Q without mentioning the limit a. Exercise 1.5.4. If a sequence (ak)k∈N converges in Q, show that (ak)k∈N is a Cauchy sequence in Q. In addition, show also that the limit a of a convergent sequence is unique. Definition 1.5.5. Let (ak)k∈N be a sequence of rational numbers. We say that (ak)k∈N is a bounded sequence if the set {ak | k ∈ N} is a bounded set in Q. Lemma 1.5.6. Let (ak)k∈N be a Cauchy sequence of rational numbers. Then (ak)k∈N is a bounded sequence. Proof. Let (ak)k∈N be a Cauchy sequence of rational numbers. Pick N ∈ N such that |an − am| 1 for n, m ≥ N. Then |an − aN| 1 for all n ≥ N, so that |an| 1 + |aN| for all n ≥ N. Let M be the max of |a1|, |a2|,..., |aN−1|, 1 + |aN|. Then (|ak|)k∈N is bounded by M. Let C denote the set of all Cauchy sequences of rational numbers. We define addition and multiplication of Cauchy sequences termwise that is, (ak)k∈N + (bk)k∈N = (ak + bk)k∈N and (ak)k∈N(bk)k∈N = (akbk)k∈N. Exercise 1.5.7. Show that the sum of two Cauchy sequences in Q is a Cauchy sequence in Q. Theorem 1.5.8. The product of two Cauchy sequences in Q is a Cauchy sequence in Q. Proof. Let (ak)k∈N and (bk)k∈N be Cauchy sequences in Q. Then |anbn − ambm| = |anbn − anbm + anbm − ambm| ≤ |an||bn − bm| + |bm||an − am| ≤ A|bn − bm| + B|an − am|, where A and B are upper bounds for the sequences (|ak|)k∈N and (|bk|)k∈N. Since (ak)k∈N and (bk)k∈N are Cauchy sequences, the theorem now follows.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2013 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.