Introduction to Mathematical Analysis I - Second Edition

22 1.5 THE COMPLETENESS AXIOM FOR THE REAL NUMBERS Since −| x | − | y | = − ( | x | + | y | ) , the conclusion follows from Proposition 1.4.2 (d). Corollary 1.4.4 For any x , y ∈ R , || x | − | y || ≤ | x − y | . Remark 1.4.5 The absolute value has a geometric interpretation when considering the numbers in an ordered field as points on a line. The number | a | denotes the distance from the number a to 0. More generally, the number d ( a , b ) = | a − b | is the distance between the points a and b . It follows easily from Proposition 1.4.2 that d ( x , y ) ≥ 0, and d ( x , y ) = 0 if and only if x = y . Moreover, the triangle inequality implies that d ( x , y ) ≤ d ( x , z )+ d ( z , y ) , for all numbers x , y , z . Exercises 1.4.1 Prove that n is an even integer if and only if n 2 is an even integer. ( Hint: prove the “if” part by contraposition, that is, prove that if n is odd, then n 2 is odd.) 1.4.2 Prove parts (c) and (d) of Proposition 1.4.1 1.4.3 Let a , b , c , d ∈ R . Suppose 0 < a < b and 0 < c < d . Prove that ac < bd . 1.4.4 Prove parts (a), (b), and (c) of Proposition 1.4.2 . 1.4.5 I Prove Corollary 1.4.4 . 1.4.6 Given two real numbers x and y , prove that max { x , y } = x + y + | x − y | 2 and min { x , y } = x + y − | x − y | 2 . 1.4.7 Let x , y , M ∈ R . Prove the following (a) | x | 2 = x 2 . (b) | x | < M if and only if x < M and − x < M . (c) | x + y | = | x | + | y | if and only if xy ≥ 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 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 .