rev311-1 - R . a. Prove that A ∪ B is a bounded set. b....

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Advanced Calculus I 311 Review Exercises 1. Prove the following assertions about a , b R . You must give reasons for every step! a. - ( a + b )= - a + - b b. - ( a/b )=( - a ) /b = a/ ( - b )if b 6 =0 c. a · a =1ifandon lyif a = - 1or a =1 d. If 0 <a< 1then0 <a n <a m < 1 for all n , m N such that n>m . 2. Prove that 5 is an irrational number. 3. Find all x R such that | x - 1 | + | x +2 | =5. 4. Let I =[0 , 1]. a. Prove that for all ±> 0, the neighborhood V ± (0) is not contained in I . b. Prove that for all a (0 , 1), there exists an ±> 0 such that V ± ( a ) is contained in I . 5. Let A and B
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Unformatted text preview: R . a. Prove that A ∪ B is a bounded set. b. Prove that sup( A ∪ B ) = sup { sup A, sup B } . 6. Let A be a nonempty bounded subset in R . Let b < 0. Let bA := { ba : a ∈ A } . Show that inf( bA ) = b (sup A ) . 7. Let S := { 1 /n-1 /m : n, m ∈ N } . Find inf S and sup S . 8. Let I n = (-1 /n, 1 /n ) for n ∈ N . Show that 0 is the only real number that belongs to all I n . 9. Prove that the set of odd numbers is countable....
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This note was uploaded on 04/02/2008 for the course MATH 311 taught by Professor Vasconceles during the Spring '08 term at Rutgers.

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