ma142ahw3 - Math 142a – Homework#3 due in class Wed Jan...

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Unformatted text preview: Math 142a – Homework #3 - due in class Wed. Jan 27 (or in drop box by 5 P.M. the next day). Read Lang, Undergraduate Analysis, Lang, Chapter 1, Sections 4, Chapter 2, Section 1, and Lecture Notes 2, pages 23-38. The homework is worth a total of 100 points, with each problem worth the same amount. ∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃ ∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃ 1) Tell whether the sequences below have a limit. Find the limit, if possible. Prove that your answer is correct. a) 1+(-1)n b) n!/nn c) (1-n)/(1+n) 2) Assume that {xn} is a sequence of real numbers. Show that if a = lim xn exists, then the set {x1,x2,x3, .... } is bounded (both above and below). n →∞ 3) Assume that {xn} and {yn} are sequences of real numbers. Show that if a = lim xn exists and n →∞ b = lim yn exists, then ab = lim xn yn exists. n →∞ n →∞ 4) True-False. State whether the following are true or false. Give a brief reason for your answer (such as a reference to some fact in Lang (including his exercises) or the lecture notes). a) Every bounded sequence of real numbers is convergent to a real number. b) Every bounded sequence of real numbers is Cauchy. c) Suppose that an ≤ bn ≤ cn for n≥n0. If the limit L = lim an = lim cn exists and is the same for n →∞ n →∞ n →∞ both {an} and {cn} , then {bn} has a limit which is also the same; i.e. L = lim bn . 5) a) Define ∞ = lim an . n →∞ b) Suppose that {an} is an unbounded sequence of non-negative real numbers. Show that {an} has a subsequence converging to ∞ according to your definition in part a). ∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃∀∂∞∀ε∃δ∃ ...
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This note was uploaded on 10/17/2010 for the course MAT Math10C taught by Professor Operadragos during the Spring '09 term at UCSD.

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