23 (c) If n>0, thenxy ≤1 2(nx 2 +1 ny 2). 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 andy, prove that max{x,y}= x+y+|x−y| 2 and min{x,y}= x+y− |x−y| 2 . 1.4.9 Let x,y,M∈R. Prove the following: (a) |x|2 =x2. (b) |x| <Mif and only if x <Mand−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) If 0≤x <ε for all ε >0, thenx =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 fields. However, we are interested in the field of real numbers. There is an additional axiom that will distinguish this ordered field from all others. In order to introduce our last axiom for the real numbers, we first need some definitions. Definition 1.5.1 Let Abe a subset of R. A number Mis called anupper bound of Aif x ≤Mfor all x ∈A. If Ahas an upper bound, thenAis said to be bounded above. Similarly, a number Lis a lower bound of Aif L≤x for all x ∈A, Ais said to be bounded belowif it has a lower bound. We also say that Ais bounded if it is both bounded above and bounded below. It follows that a set Ais bounded if and only if there exist M∈Rsuch that |x| ≤Mfor all x ∈A (see Exercise 1.5.1). The following concept plays a central role in the study of the real numbers. Definition 1.5.2 — Supremum of a set. Let Abe a nonempty set that is bounded above. We call a number α a least upper bound or supremumof A, if the following conditions hold: (1) x ≤α for all x ∈A(that is, α is an upper bound of A). (2) If Mis an upper bound of A, then α≤M(this means α is smallest among all upper bounds). It can be shown that if Ahas 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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