04fall-test1

# 04fall-test1 - MAT 371 Advanced Calculus September 30, 2004...

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MAT 371 Advanced Calculus /120 September 30, 2004 Test 1 name 1 2 3 4 5 6 15 15 20 30 20 20 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 1. a. State the least upper bound axiom (or supremum axiom ). b. State “the” (“a” ?) deﬁnition of limit point . c. State “the” (“a” ?) deﬁnition of open set . 2. a. Using only the ﬁeld axioms, prove ONE of the following: (i) x R , 0 · x = 0 , or (ii) x R , ( - 1) · x = - x . If impossible explain why. b. Using only the ﬁeld axioms, prove that - 1 6 = 1 If impossible explain why. 3. Suppose that { I i = [ a i , b i ]: i Z + } is a collection of nested nontrivial intervals in R , i.e., for all i Z + , a i a i +1 < b i +1 b i . Prove that the intersection T i =1 I i is nonempty. Bonus Consider the sequences ( a n ) n =1 and ( b n ) n =1 of rational (!) numbers deﬁned by a n = (1 + 1 n ) n and b n = (1 + 1 n ) n +1 . Show that ( a n ) n =1 and ( b n ) n =1 are monotonically increasing and decreasing, respectively. (Hint: Analyze the quotients a n +1 a n and b n +1 b n .) Also show that both sequences are bounded. What can you conclude about the convergence of the sequences? 4. a. Suppose that S R is a subset of the set of real numbers with supremum z = sup S .
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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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