26 1.5 The Completeness Axiom for the Real Numbers The following proposition is useful when dealing with infma and its proof is completely analogous to that of Proposition 1.5.1. Proposition 1.5.2 Let A be a nonempty subset of R that is bounded below. Then β = infA if and only if (1’) x ≥ β for all x ∈ A, (2’) For any ε > 0, there exists a ∈ A such 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 M be an upper bound for B, then for x ∈ B, x ≤ M. In particular, it is also true that x ≤ M for x ∈ A since A ⊂ B. Thus, A is also bounded above. Now, since supB is an upper bound for B, it is also an upper bound for A. Then, by the second condition in the defnition of supremum, supA ≤ supB as desired. □ It will be convenient for the study of limits of sequences and functions to introduce two additional symbols. Defnition 1.5.4 The extended real number system consists of the real feld R and the two symbols ∞ and −∞. We preserve the original order in R and defne −∞ < x < ∞ for every x ∈ R The extended real number system does not form an ordered feld, but it is customary to make the following conventions: (a) If x is a real number, then x + ∞ = ∞, x +(−∞)= −∞. (b) If x > 0, then x · ∞ = ∞, x · (−∞)= −∞. (c) If x < 0, then x· ∞ = −∞, x · (−∞)= ∞. (d) ∞ + ∞ = ∞, −∞ +(−∞)= −∞, ∞ · ∞ =(−∞) · (−∞)= ∞, and (−∞) · ∞ = ∞ · (−∞)= −∞. We denote the extended real number set by R. The expressions 0 · ∞, ∞ +(−∞), and (−∞)+ ∞ are left undefned. The set R with the above conventions will be convenient when describing results about limits in later chapters. Defnition 1.5.5 If A ≠ 0/ is not bounded above in R, we will write supA = ∞. If A is not bounded below in R, we will write infA = −∞. With this defnition, every nonempty subset of R has a supremum and an infmum 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 A of R is bounded if and only if there is M ∈ R such that |x|≤ M for all x ∈ A.
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