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# tut01 - The University of Sydney School of Mathematics and...

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Unformatted text preview: The University of Sydney School of Mathematics and Statistics 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 } (b) B = { 1 − 1 /n : n ∈ N } (c) C = { (1 / 2 n ): n ∈ N } 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 ); (b) sup( A + B ) = sup A + sup B ; (c) If A ⊂ B then sup A ≤ sup B ; *(d) If s := sup A negationslash∈ A , then there exist strictly increasing x n ∈ A with sup n ∈ N x n = s . 3. Let x k > 0 for k = 1 ,... ,n . Use the Cauchy Schwarz inequality to prove that n 2 ≤ parenleftBig n summationdisplay...
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