26 1.5 The Completeness Axiom for the Real Numbers The following proposition is useful when dealing with infima and its proof is completely analogous to that of Proposition 1.5.1. Proposition 1.5.2 Let Abe a nonempty subset of Rthat is bounded below. Thenβ =infAif and only if (1’) x ≥β for all x ∈A, (2’) For any ε >0, there exists a∈Asuch that a<β+ε. The following is a basic property of suprema. Additional ones are described in the exercises. Theorem 1.5.3 Let A and B be nonempty sets and A⊂B. Suppose B is bounded above. Then supA≤supB. Proof: Let Mbe an upper bound for B, then for x ∈B, x ≤M. In particular, it is also true that x ≤Mfor x ∈Asince A⊂B. Thus, Ais also bounded above. Now, since supBis an upper bound for B, it is also an upper bound for A. Then, by the second condition in the definition of supremum, supA≤supBas desired. □ It will be convenient for the study of limits of sequences and functions to introduce two additional symbols. Definition 1.5.4 The extended real number systemconsists of the real fieldRand the two symbols ∞and−∞. We preserve the original order inRand define −∞<x <∞for every x ∈R The extended real number system does not form an ordered field, but it is customary to make the following conventions: (a) If x is a real number, thenx+∞=∞, x+(−∞) =−∞. (b) If x >0, thenx· ∞=∞, x· (−∞) =−∞. (c) If x <0, thenx· ∞=−∞, x· (−∞) =∞. (d) ∞+∞=∞, −∞+(−∞) =−∞, ∞· ∞= (−∞)· (−∞) =∞, and (−∞)· ∞=∞· (−∞) =−∞. We denote the extended real number set by R. The expressions 0· ∞, ∞+(−∞), and(−∞)+∞ are left undefined. The set Rwith the above conventions will be convenient when describing results about limits in later chapters. Definition 1.5.5 If A̸=0/ is not bounded above inR, we will write supA=∞. If Ais not bounded below in R, we will write infA=−∞. With this definition, every nonempty subset of Rhas a supremum and an infimum in R. To complete the picture we adopt the following conventions for the empty set: sup 0/ =−∞ and inf 0/ =∞. Exercises 1.5.1 Prove that a subset Aof Ris bounded if and only if there is M∈Rsuch that |x| ≤Mfor all x ∈A.
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