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Ma28_S08_Exam1_soln

# Ma28_S08_Exam1_soln - Math 28 Spring 2008 Exam 1...

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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 ² > 0, then by the characterization of supremum, there exists an a ∈ A such that s- ² < a . Since s 6∈ A we have a 6 = s and hence V ² ( 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 ² > ∃ N 1 ∈ N such that for all n ≥ N 1 we have | b n- b | < ² 2 . So we also have that for all n ≥ N 1...
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Ma28_S08_Exam1_soln - Math 28 Spring 2008 Exam 1...

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