05sprg-test1

# 05sprg-test1 - from the deﬁnition of “ convergence”...

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MAT 371 Advanced Calculus /120 February 16, 2005 Test 1 name 1 2 3 4 5 6 15 15 25 15 25 25 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 1. a. Deﬁne upper bound and least upper bound . b. State the least upper bound axiom (or supremum axiom ). c. Deﬁne open ball and open set . 2. Using only the ﬁeld axioms, prove the following (if impossible explain why): a. x, y R , if x · y = 0 then x = 0 or y = 0, and b. x, y R , if x 6 = 0 and y 6 = 0 then ( xy ) - 1 exists, and ( xy ) - 1 = x - 1 y - 1 . 3. a. Show that if a set S R has a least upper bound, then the least upper bound is unique. b. Give a detailed outline of a proof that there exists a real number z such that z 2 = 3. 4. a. Prove that the intersection of two open sets is open. b. Prove that the union of any collection of open sets is open. c. Give an example of a countable collection of open intervals whose intersection is the closed interval [0 , 1]. 5. Suppose that ( a n ) n =1 and ( b n ) n =1 are converging sequences of real numbers. Working
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Unformatted text preview: from the deﬁnition of “ convergence” prove that the sequence (3 a n-2 b n ) ∞ n =1 converges. 6. Suppose that ( a n ) ∞ n =1 and ( b n ) ∞ n =1 are sequences of real numbers that converge to real numbers L and M , respectively. a. Prove that if for all n ∈ Z + a n < b n , then L ≤ M . b. Give an explicit counterexample that shows that (under the same hypotheses as above) the conclusion L ≤ M cannot be replaced by L < M . c. Conversely suppose that ( a n ) ∞ n =1 and ( b n ) ∞ n =1 are sequences of real numbers that converge to real numbers L and M , respectively. What can you say (without proof) about the relationship between a n and b n if you know only that L < M ?...
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## This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

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