# tut01s - The University of Sydney School of Mathematics and...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the 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 non-empty 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

{[ snackBarMessage ]}

### Page1 / 3

tut01s - The University of Sydney School of Mathematics and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online