practiceexam1

# practiceexam1 - , that M is bounded below, and that sup S =...

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MATH 131A - Practice Midterm 1. Please write clearly, and show your reasoning with mathematical rigor. You may use any correct rule about the algebra or order structure of R from Section 3 without proving it. 1. a) State the Principle of Mathematical Induction. b) Prove that for all positive integers n , 1 + 3 + ... + (2 n - 1) = n 2 . 2. a) State the Least Upper Bound Axiom (also called the Completeness Axiom). b) Let S be a non-empty ( S 6 = ) subset of the real numbers such that S is bounded above. Let M = { y : y x for all x S. } . Prove that M 6 =
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Unformatted text preview: , that M is bounded below, and that sup S = inf M. 3. Let-A be a negative real number. Use the Least Upper Bound Axiom to prove that there is an integer (necessarily negative) n such that n <-A. 4. a) Give the ±-N deﬁnition of lim n →∞ s n = s. b) Use the deﬁnition to prove that lim n →∞ 10 n +(-1) n n = 10 . 5. Carefully prove that if lim n →∞ a n = a and lim n →∞ b n = b , where a and b are real numbers, then lim n →∞ ( a n + b n ) = a + b. 1...
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## This note was uploaded on 11/15/2010 for the course MATH math 131 taught by Professor Kim during the Spring '10 term at UCLA.

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