Sample Exam 1D

Sample Exam 1D - where . Prove that . 2. Show that 3. Prove...

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MATH 311 EXAM 1D Directions for Part I: For each of the following, provide the appropriate definition or theorem. 1. Bounded Set 2. Least Upper Bound 3. Axiom of Completeness 4. Convergent Sequence 5. Cauchy Sequence 6. Subsequence 7. Monotone Convergence Theorem 8. Nested Interval Theorem 9. Bolzano-Weierstrass Theorem 10. Cauchy Criterion Directions for Part II: Complete the following problems. If a problem asks for a proof, make explicit what you are assuming. Make sure you justify each step. Good luck! 1. Let p be a positive real number with . Suppose
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Unformatted text preview: where . Prove that . 2. Show that 3. Prove the Monotone Convergence Theorem. 4. Prove for some constant , then . 5. Suppose . Consider the sequence . Does this sequence converge? Does the sequence diverge? Will it converge for some values of a and b but not for others? 6. Prove the sequence defined by and for all n converges. 7. Use the Nested Interval Theorem to prove the Bolzano-Weierstrass Theorem 8. Prove that is a sequence converges to some limit L , then each of its subsequences also converges to L ....
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