27 1.5.2 Let A be a nonempty set and suppose α1 and α2 satisfy conditions (1) and (2) in Definition 1.5.2 (that is, both are suprema of A). Prove that α1 =α2. 1.5.3 For each subset of Rbelow, determine if it is bounded above, bounded below, or both. If it is bounded above (below) find the supremum (infimum). Justify all your conclusions. (a) {1,5,17} (b) [0,5) (c) 1+ (−1)n 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 AandBare nonempty subsets of Rthat are bounded above. Define A+B={a+b: a∈Aandb∈B}. Prove that A+Bis bounded above and sup(A+B) =supA+supB. 1.5.5 Let Abe a nonempty subset of R. Define −A={−a: a∈A}. (a) Prove that if Ais bounded below, then −Ais bounded above. (b) Prove that if Ais bounded below, then Ahas an infimum inRand infA=−sup(−A). 1.5.6 Let Abe a nonempty subset of Randt ∈R. Define tA={ta: a∈A}. Prove the following statements: (a) If t >0 and Ais bounded above, thentAis bounded above and sup(tA) =t supA. (b) If t <0 and Ais bounded above, thentAis bounded below and inf(tA) =t supA. 1.5.7 Suppose Aand Bare nonempty subsets of Rthat are bounded below. Prove that A+Bis bounded below and inf(A+B) =infA+infB. 1.5.8 Let A,Bbe nonempty subsets of Rthat 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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