Sample Exam 1C - the set of positive integers is not...

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MATH 311 EXAM 1C 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. Prove that for every real number x , there is a positive integer n such that ; that is, prove that
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Unformatted text preview: the set of positive integers is not bounded above. Furthermore, use this result to prove that for any , there is some positive integer n such that . 2. Prove that for real numbers with , then there is a rational number x such that . 3. Verify with an proof that 4. Suppose and for all n . Prove directly that if and are Cauchy, then is also Cauchy. 5. For each positive integer n , set Prove that . You may use the fact that the sum of the first n positive integers is 6. Prove the Cauchy Criterion: A sequence converges if and only if it is Cauchy....
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