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Introductory Real Analysis University of Washington Math 327, Autumn 2018 c 2018, Dr. F. Dos Reis Homework Exercise 1. (3points) Let ( F, + , * ) be a commutative field and a an element of F . Prove that ( a - 1 ) - 1 = a . This is proposition 2.8. You may use any result stated above proposition 2.8 but not the proposition 2.8. 1
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Introductory Real Analysis Math 327, Autumn 2018 University of Washington c 2018, Dr. F. Dos Reis Exercise 2. (4 points) Let ( F, + , * ) a commutative field totally ordered. 1. (Archimedean Property) For any two positive elements of F , a and b , there exists n N such that na > b . 2. (Proposition 2) For any element z > 0 in F , there exists n N such that 1 n < z . Prove that the Archimedean property is true in F iff Proposition 2 is true in F . 2
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Introductory Real Analysis Math 327, Autumn 2018 University of Washington c 2018, Dr. F. Dos Reis Exercise 3. (4 points) Let u n = 3 n + 2 4 n - 2 + r 9 n n 2 + 1 . Prove using the definition of the limit of u n that lim n →∞ u n = 3 4 . 3
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Introductory Real Analysis Math 327, Autumn 2018 University of Washington c 2018, Dr. F. Dos Reis Exercise 4. (3 points) Let u n be a non decreasing sequence ( u n +1 > u n ). Prove that if u n is unbounded, then lim n →∞ u n = 4
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Introductory Real Analysis Math 327, Autumn 2018 University of Washington c 2018, Dr. F. Dos Reis Exercise 5. (3 points) Prove Theorem 38 (Comparison test) Let a n and b n series with non negative terms such that for all the index greater than some N , 0 6 a n 6 b n . If b n is convergent, then a n is convergent. If a n is divergent, then b n is divergent. You may use any result stated prior to theorem 38 but not theorem 38. 5
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Introductory Real Analysis Math 327, Autumn 2018 University of Washington c 2018, Dr. F. Dos Reis Exercise 6. (4 points) Determine whether the following sequences are convergent, divergent, absolutely con- vergent, conditionally convergent: 1. X n =1 ( - 1) n 2 n + cos n + 5 . 2. X n =1 p (2 n )! n ! . 6
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Introductory Real Analysis Math 327, Autumn 2018 University of Washington c 2018, Dr. F. Dos Reis Exercise 7. (3 points) Let f be a continuous function on (1 , 2) such that lim x 1 f ( x ) < lim x 2 f ( x ). Prove that for any c between lim x 1 f ( x ) and lim x 2 f ( x ), there exists x 0 (1 , 2) such that f ( x 0 ) = c . 7
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Introductory Real Analysis Math 327, Autumn 2018 University of Washington c 2018, Dr. F. Dos Reis Exercise 8. ( 3 points) Let f be a uniformly continuous function on an interval I . Let g be a uniformly continuous function on an interval J . Assume that f ( I ) J . Prove that the composition g ( f ( x )) is a uniformly continuous on I . Remark: f ( I ) is the set of all the images of elements of I : f ( I ) = { f ( x ) , x I } . 8
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Introductory Real Analysis Math 327, Autumn 2018 University of Washington c 2018, Dr. F. Dos Reis 1 Axioms of R Definition 1. ( F, + , * ) is a commutative field if 1. For any a , b , c in F , a + ( b + c ) = ( a + b ) + c , and a * ( b * c ) = ( a * b ) * c (associativity).
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