Sample Exam 1A

Sample Exam 1A - 2 Give an example of each of the following...

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MATH 311 EXAM 1A 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. Suppose S and T are nonempty subsets of R that are bounded above. a. Explain why and exist. b. Show that where . c. Is it necessarily true that where ? Either prove or provide a counterexample.
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Unformatted text preview: 2. Give an example of each of the following, or argue that such a request is impossible. a. A Cauchy sequence that is not monotone b. A monotone sequence that is not Cauchy c. A Cauchy sequence with a divergent subsequence d. An unbounded sequence containing a subsequence that is Cauchy 3. Give an proof to verify that 4. Does the sequence defined by converge (to a finite number)? Either find its limit and verify it, or prove that the limit does not exist. 5. Let and let for all positive integers n . Prove converges. 6. Prove that every bounded sequence has a convergent subsequence. \...
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