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Sample Exam 2C

Sample Exam 2C - 1 Prove that p is an accumulation point of...

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MATH 311 EXAM 2C Directions for Part I: For each of the following, provide the appropriate definition or theorem. 1. Accumulation Point 2. Functional Limit 3. Monotone Function 4. Continuity at a Point 5. Uniform Continuity 6. Open Set and Closed Set 7. Compact Set 8. Heine-Borel Theorem 9. Extreme Value Theorem 10. Intermediate Value Theorem 11. *1 st Continuous Function Theorem 12. *2 nd Continuous Function Theorem 13. *4 th Continuous Function Theorem 14. *Sequential Compactness Theorem 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!
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Unformatted text preview: 1. Prove that p is an accumulation point of the set E if and only if there exists a sequence of distinct points in such that 2. Verify that 3. Suppose is continuous and for all numbers of the form , , it follows that . Prove that for all x . 4. Prove that given by is uniformly continuous on but not on . 5. Prove directly (Do not use Heine-Borel): If K is compact, F is closed, and , then is compact. 6. Suppose K is compact. Prove that every sequence of points in K has a convergent subsequence whose limit is in K ....
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