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Introductory Real Analysis University of Washington Math 327, Spring 2014 c 2014, Dr. F. Dos Reis Last Name (PRINT): First Name (PRINT): Spring 2014 – Introductory Real Analysis Final Examination Instructions 1. The use of all electronic devices is prohibited. 2. Present your solutions in the space provided. Show all your work neatly and concisely. Clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. Scholastic dishonesty will not be tolerated. The work on this test is my own. Signature: 1
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Introductory Real Analysis Math 327, Spring 2014 University of Washington c 2014, Dr. F. Dos Reis Questions 1 2 3 4 5 6 7 8 9 Total Grade: Exercise 1. (12 point) Prove the following proposition: The additive inverse is unique. This result is Proposition 2.1. You may use for the proof any results mentioned PRIOR to Proposition 2.1 For any real number a , - a = ( - 1) * a . This result is Proposition 2.5. You may use for the proof any results mentioned PRIOR to Proposition 2.5 2
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Introductory Real Analysis Math 327, Spring 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 2. (10 points) Prove proposition 9: The cut number is unique. You may use any result prior to Proposition 9. You may try a proof by contradiction and assuming that there exists 2 different cut numbers. 3
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Introductory Real Analysis Math 327, Spring 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 3. (12 points) Prove that if A is a non empty subset of B , inf( B ) 6 inf( A ) 6 sup( A ) 6 sup( B ) . You may use any result prior to Proposition 25. 4
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Introductory Real Analysis Math 327, Spring 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 4. (10 points) Use the definition of limits (Definition 27 with ) to prove that lim n →∞ n sin n n 2 + 1 = 0 5
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Introductory Real Analysis Math 327, Spring 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 5. (12 points) Let S be the set of rational numbers r such that 0 < r < 1, is the set S open? is it closed? Justify your answer. 6
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Introductory Real Analysis Math 327, Spring 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 6. (8 points) Let S be a non empty subset of the real numbers. Assume that L is the greatest lower bound of S and L is not in S , Prove that L is an accumulation point of S . 7
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Introductory Real Analysis Math 327, Spring 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 7. (14 points) Determine whether the following series are convergent, absolutely convergent, conditionally convergent. 1. S = X n =1 ( n + 1)( n + 2) 2 n n 2 2. T = X n =1 ( - 1) n n 1 + n 8
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Introductory Real Analysis Math 327, Spring 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 8. (15 points) Given the sequence of functions f n ( x ) = n + cos( nx ) 2 n + 1 1. Is f n convergent? 2. Is f n uniformly convergent on R ? Justify your answer.
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  • Fall '08
  • Staff
  • Math, Introductory Real Analysis, Dr. F. Dos Reis, Dr. F. Dos

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