Analysis Extra Exercises 3 PDF

Preview

Summary

N/A...

Description

Additional Exercises sheet 3: Supremum and Infimum 1. Let d > 1 and M > 0. Prove that there exists n ∈ N such that dn > M . 2. Find the supremum and infimum for each of the following subsets of R. (a) A1 = {x ∈ R : |2x − 1| < 11}; (b) A2 = {x + |x − 1| : x ∈ R};   (c) A3 = 1 − 1/n : n ∈ Z\{0} ;

(d) A4 = {2−m + 3−n + 5−p : m, n, p ∈ N};   (−1)n n (e) A5 = :n∈N . 2n + 1 3. Let A=



  n  2 −1 1 3 7 15 , , , ,... = :n∈N . 2 4 8 16 2n

Prove that A is bounded and that sup A = 1. 4. Suppose that A and B are non-empty subsets of [0, ∞) that are bounded above. Denote A · B = {ab : a ∈ A and b ∈ B}. Show that sup(A · B) = (sup A)(sup B ).

1...