27 1.5.2 Let A be a nonempty set and suppose α1 and α2 satisfy conditions (1) and (2) in Defnition 1.5.2 (that is, both are suprema of A). Prove that α1 = α2. 1.5.3 For each subset of R below, determine if it is bounded above, bounded below, or both. If it is bounded above (below) fnd the supremum (infmum). Justify all your conclusions. (a) {1,5,17} (b) [0,5) (−1)n (c) 1+ : n ∈ N n (d) (−3,∞) (e) {x ∈ R : x2 − 3x + 2 = 0} (f) {x2 − 3x + 2 : x ∈ R} (g) {x ∈ R : x3 − 4x < 0} (h) {x ∈ R : 1 ≤|x| < 3} 1.5.4 ▶ Suppose A and B are nonempty subsets of R that are bounded above. Defne A + B = {a + b : a ∈ A and b ∈ B}. Prove that A + B is bounded above and sup(A + B)= supA+ supB. 1.5.5 Let A be a nonempty subset of R. Defne −A = {−a : a ∈ A}. (a) Prove that if A is bounded below, then −A is bounded above. (b) Prove that if A is bounded below, then A has an infmum in R and infA = −sup(−A). 1.5.6 Let A be a nonempty subset of R and t ∈ R. Defne tA = {ta : a ∈ A}. Prove the following statements: (a) If t > 0 and A is bounded above, then tA is bounded above and sup(tA)= t supA. (b) If t < 0 and A is bounded above, then tA is bounded below and inf(tA)= t supA. 1.5.7 Suppose A and B are nonempty subsets of R that are bounded below. Prove that A + B is bounded below and inf(A + B)= infA + infB. 1.5.8 Let A,B be nonempty subsets of R that are bounded below. Prove that if A ⊂ B, then infA ≥ infB. 1.6 Applications of the Completeness Axiom We prove here several fundamental properties of the real numbers that are direct consequences of the Completeness Axiom. Theorem 1.6.1 — The Archimedean Property. The set of natural numbers is unbounded above.
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