This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 1 (Week 2) MATH2962: Real and Complex Analysis (Advanced) Semester 1, 2009 Web Page: http://www.maths.usyd.edu.au/u/UG/IM/MATH2962/ Lecturer: Daniel Daners Questions marked with * are more difficult questions. Questions to complete during the tutorial 1. Determine the supremum of the following sets. Determine whether it is a maximum or not. (a) A = { x ∈ Q : x 2 ≤ 8 } Solution: sup A = 2 √ 2 is not a maximum because 2 √ 2 negationslash∈ Q . (b) B = { 1 − 1 /n : n ∈ N } Solution: sup B = 1 is not a maximum as 1 − 1 /n < 1 for all n ∈ N (c) C = { (1 / 2 n ): n ∈ N } Solution: sup C = 1 = max C since (1 / 2) = 1 is an element of C . 2. Suppose A,B are nonempty subsets of R . We set A + B := { x + y : x ∈ A,y ∈ B } and − A := {− x : x ∈ A } . Prove the following statements. (a) sup( A ) = − inf( − A ); Solution: Set L := inf( − A ). We want to show that sup A = − L . We first show that − L is an upper bound for A . Since L = inf( − A ) we have L ≤ − x for all x ∈ A and so x ≤ − L for all x ∈ A . Hence − L is an upper bound for A . Next we need to show that given an arbitrary upper bound m of A , it follows that − L ≤ m . Since m is an upper bound for A we have x ≤ m and so − m ≤ − x for all x ∈ A . (b) sup( A + B ) = sup A + sup B ; Solution: The idea is to prove two inequalities: sup( A + B ) ≤ sup A + sup B , and sup( A + B ) ≥ sup A +sup B . Since sup A is an upper bound for A we have x ≤ sup A for all x ∈ A . Similarly, y ≤ sup B for all y ∈ B . Hence x + y ≤ sup A + sup B for all x ∈ A and y ∈ B , that is, sup A + sup B is an upper bound for A + B . Hence by definition of a supremum sup( A + B ) ≤ sup A + sup B . We next show the opposite inequality. By....
View
Full Document
 Three '09
 daners
 Statistics, Inequalities, Ri, sup, Cauchy–Schwarz inequality, Hölder's inequality

Click to edit the document details