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Sample Exam 1B - has no rational least upper bound 4 Verify...

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MATH 311 EXAM 1B 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 is a nonempty subset of R that is bounded below. Prove that S has a greatest lower bound in R . 2. Prove that is irrational. 3. Let represent the set of positive rational numbers. Show that the set
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Unformatted text preview: has no rational least upper bound. 4. Verify with an proof that 5. We say a sequence is almost Cauchy or pseudo-Cauchy if for every , there is some positive integer N such that for all , . Is it true that every pseudo-Cauchy sequence is convergent? If so, prove it. If not, provide a counterexample. Is it true that every convergent sequence is pseudo-Cauchy? If so, prove it. If not, provide a counterexample. 6. Let S be nonempty set of real numbers that is bounded above. Let . If , prove that there exists a sequence of terms completely in S such that for each index n ....
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