Introduction to Mathematical Analysis I - Second Edition

24 1.5 THE COMPLETENESS AXIOM FOR THE REAL NUMBERS an upper bound of A but M < 4. Choose a number y ∈ R such that M < y < 4 and 0 < y . Set x = − √ y . Then − 2 < x < 0 < 1 and, so, y = x 2 ∈ A . However, y > M which contradicts the fact that M is an upper bound. Thus 4 ≤ M . This proves that 4 = sup A . The following proposition is convenient in working with suprema. Proposition 1.5.1 Let A be a nonempty subset of R that is bounded above. Then α = sup A if and only if (1’) x ≤ α for all x ∈ A ; (2’) For any ε > 0, there exists a ∈ A such that α − ε < a . Proof: Suppose first that α = sup A . Then clearly (1’) holds (since this is identical to condition (1) in the definition of supremum). Now let ε > 0. Since α − ε < α , condition (2) in the definition of supremum implies that α − ε is not an upper bound of A . Therefore, there must exist an element a of A such that α − ε < a as desired. Conversely, suppose conditions (1’) and (2’) hold. Then all we need to show is that condition (2) in the definition of supremum holds. Let M be an upper bound of A and assume, by way of con- tradiction, that M < α . Set ε = α − M . By condition (2) in the statement, there is a ∈ A such that a > α − ε = M . This contradicts the fact that M is an upper bound. The conclusion now follows. The following is an axiom of the real numbers and is called the completeness axiom . The Completeness Axiom. Every nonempty subset A of R that is bounded above has a least upper bound. That is, sup A exists and is a real number. This axiom distinguishes the real numbers from all other ordered fields and it is crucial in the proofs of the central theorems of analysis. There is a corresponding definition for the infimum of a set. Definition 1.5.3 Let A be a nonempty subset of R that is bounded below. We call a number β a greatest lower bound or infimum of A , denoted by β = inf A , if (1) x ≥ β for all x ∈ A (that is, β is a lower bound of A ); (2) If N is a lower bound of A , then β ≥ N (this means β is largest among all lower bounds). Using the completeness axiom, we can prove that if a nonempty set is bounded below, then its infimum exists (see Exercise 1.5.5 ) . Example 1.5.2 (a) inf ( 0 , 3 ] = inf [ 0 , 3 ] = 0. (b) inf { 3 , 5 , 7 , 8 , 10 } = 3. (c) inf ( − 1 ) n n : n ∈ N = − 1. (d) inf { 1 + 1 n : n ∈ N } = 1. (e) inf { x 2 : − 2 < x < 1 , x ∈ R } = 0.