MAT321-Lecture Notes A. TUFA Exercise 1.1 1. Determine whether the following sets are bounded from below, above, or both. If so, determine their infimum and/or supremum and find out whether these infima/suprema are actually minima/maxima. a) {x ∈ R : x2 + x + 3 > 0}. b) {2 − (−1)n : n ∈ N}. 2. Find sup A and inf A if any and prove that your answer is correct. a) A = {2, 3, 16, 19.007, 23}. 1 1 1 b) A = 1, , , · · · = :n∈N . 2 3 n c) A = {x ∈ R : x2 < 7}. d) A = {x ∈ Q : 0 ≤ x ≤ √ 2}. 3. Let A = (−1, 2]. Prove that inf A = −1, but min A doesn’t exist. 4. If a ∈ R and a < 0, then prove that sup(aA) = a inf A, where aA = {ax : x ∈ A}. 5. Let A and B nonempty subsets of R which are both bounded from above. Define the set A + B as A + B = {a + b : a ∈ A, b ∈ B}. Show that sup(A + B) = sup A + sup B. 6. Let {an } be a sequence defined by the following relations. √ √ a1 = 2, an+1 = 2 + an , ∀n > 1. Use the principle of mathematical induction to show that a) an ≤ 2 for all n. a) an ≤ an+1 for all n. 7. Prove that Q∗ is dense in R. 9