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Unformatted text preview: Math 28 Spring 2008: Exam 1 Instructions: Each problem is scored out of 10 points for a total of 50 points. You may not use any outside materials(eg. notes or calculators). You have 50 minutes to complete this exam. Problem 1. Let A R be bounded above. Show that sup( A ) A . Proof. Let A be bounded above and let s = sup( A ) which exists by the completeness of R . Then we need to see that s A = A L where L is the set of limit points of A . Since s can either be in A or not in A , we examine each case separately. If s A then clearly s A . If s 6 A , then we will show that s is a limit point. Let > 0, then by the characterization of supremum, there exists an a A such that s- < a . Since s 6 A we have a 6 = s and hence V ( s ) intersects A in a point not s . Therefore, s is a limit point of A and hence s A . Problem 2. Let ( a n ) a and ( b n ) b be convergent sequences. Show directly that ( a n- b n ) a- b . (i.e. without using the Algebraic Limit theorem). Proof. We know that ( b n ) b and hence for all > N 1 N such that for all n N 1 we have | b n- b | < 2 . So we also have that for all n N 1...
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This note was uploaded on 03/02/2010 for the course MATH 28 taught by Professor Staff during the Spring '08 term at UMass (Amherst).
- Spring '08