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Ma28_S08_Exam1

# Ma28_S08_Exam1 - Theorem(b Show that the union of ﬁnitely...

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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 . 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). Problem 3. (a) State the definition of a compact set and the characterization of compact sets by the Heine-Borel
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Unformatted text preview: Theorem. (b) Show that the union of ﬁnitely many compact sets is compact. Problem 4. Prove the ∑ ∞ n =1 a n is convergent if and only if for any m ∈ N ∑ ∞ n =1 a m + n is convergent. Moreover, ∑ ∞ n =1 a n converges to a 1 + ··· + a m + ∑ ∞ n =1 a n + m . Problem 5. (a) State the Monotone Converge Theorem. (b) Deﬁne a 1 = 1 and a n = 3-1 a n-1 for n ≥ 2. Determine the convergence or divergence of the sequence ( a n ). 1...
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