Introduction to Mathematical Analysis I - Second Edition

25 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 A be a nonempty subset of R that is bounded below. Then β = inf A 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 sup A ≤ sup B . 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 . Thus, A is also bounded above. Now, since sup B is an upper bound for B , it is also an upper bound for A . Then, by the second condition in the definition of supremum, sup A ≤ sup B as 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 system consists of the real field R and the two symbols ∞ and − ∞ . We preserve the original order in R and 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, 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 undefined. The set R with the above conventions will be convenient to describe results about limits in later chapters. Definition 1.5.5 If A 6 = /0 is not bounded above in R , we will write sup A = ∞ . If A is not bounded below in R , we will write inf A = − ∞ . With this definition, every nonempty subset of R has 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 A of R is bounded if and only if there is M ∈ R such that | x | ≤ M for all x ∈ A .