23 2 1 (c) If n > 0, then xy ≤ 1 2 (nx + y 2). n 1.4.6 Prove parts (i), (ii), and (iii) of Proposition 1.4.2. 1.4.7 ▶ Prove Corollary 1.4.4. 1.4.8 Given two real numbers x and y, prove that x + y+ |x − y| x + y−|x − y| max{x,y} = and min{x,y} = . 2 2 1.4.9 Let x,y,M ∈ R. Prove the following: 2 (a) |x|2 = x . (b) |x| < M if and only if x < M and −x < M. (c) |x + y| = |x| + |y| if and only if xy ≥ 0. 1.4.10 Let x,y,z ∈ R. Prove the following statements. (a) If0 ≤ x < ε for all ε > 0, then x = 0. (b) The following are equivalent: (i) y ≤ z. (ii) y < z+ ε for all ε > 0. 1.5 The Completeness Axiom for the Real Numbers There are many examples of ordered felds. However, we are interested in the feld of real numbers. There is an additional axiom that will distinguish this ordered feld from all others. In order to introduce our last axiom for the real numbers, we frst need some defnitions. Defnition 1.5.1 Let A be a subset of R. A number M is called an upper bound of A if x ≤ M for all x ∈ A. If A has an upper bound, then A is said to be bounded above. Similarly, a number L is a lower bound of A if L ≤ x for all x ∈ A, A is said to be bounded below if it has a lower bound. We also say that A is bounded if it is both bounded above and bounded below. It follows that a set A is bounded if and only if there exist M ∈ R such that |x|≤ M for all x ∈ A (see Exercise 1.5.1). The following concept plays a central role in the study of the real numbers. Defnition 1.5.2 — Supremum of a set. Let A be a nonempty set that is bounded above. We call a number α a least upper bound or supremum of A, if the following conditions hold: (1) x ≤ α for all x ∈ A (that is, α is an upper bound of A). (2) If M is an upper bound of A, then α ≤ M (this means α is smallest among all upper bounds). It can be shown that if A has a supremum, then it has only one (see Exercise 1.5.2). In this case, we denote such a number by supA.
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