Introduction to Mathematical Analysis I - Second Edition

26 1.6 APPLICATIONS OF THE COMPLETENESS AXIOM 1.5.2 Let A be a nonempty set and suppose α and β satisfy conditions (1) and (2) in Definition 1.5.2 (that is, both are suprema of A ). Prove that α = β . 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) 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 : x 2 − 3 x + 2 = 0 } (f) { x 2 − 3 x + 2 : x ∈ R } (g) { x ∈ R : x 3 − 4 x < 0 } (h) { x ∈ R : 1 ≤ | x | < 3 } 1.5.4 I Suppose A and B are nonempty subsets of R that are bounded above. Define A + B = { a + b : a ∈ A and b ∈ B } . Prove that A + B is bounded above and sup ( A + B ) = sup A + sup B . 1.5.5 Let A be a nonempty subset of R . Define − 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 infimum in R and inf A = − sup ( − A ) . 1.5.6 Let A be a nonempty subset of R and α ∈ R . Define α A = { α a : a ∈ A } . Prove the following statements: (a) If α > 0 and A is bounded above, then α A is bounded above and sup α A = α sup A . (b) If α < 0 and A is bounded above, then α A is bounded below and inf α A = α sup A . 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 ) = inf A + inf B . 1.5.8 Let A , B be nonempty subsets of R that are bounded below. Prove that if A ⊂ B , then inf A ≥ inf B . 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.