Fall 2010 Midterm Solutions

Fall 2010 Midterm Solutions - R then inf E is an element of...

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Math 140A, Fall 2010, Midterm, 11/8/10 Instructions . Answer all questions. You may use without proof anything which was proved in class. Cite a theorem either by name, if it has one, or by brieFy stating what it says. 1. (20 points) Give an example of an open cover of the interval (0 , 1) R which has no ±nite subcover. Solution . Let U i = (1 /i, 1), i 2. Then { U i } is a cover of (0 , 1) with no ±nite subcover. 2. (10 points each). True or false? ²or each one, give a brief reason (a complete proof is not necessary) if true, and a counterexample or brief reason as appropriate if false. (a) The set of irrational real numbers is uncountable. Solution . True: R is uncountable and Q is countable, hence R \ Q is still uncountable; otherwise R = ( R \ Q ) Q would be countable. (b) Let E R be the set of rational numbers x such that 0 < x < 1. Is 1 / 2 an interior point of E ? Solution . No: any ball centered at 1 / 2 necessarily contains an irrational number, which won’t be in E . (c) If E is a bounded subset of
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Unformatted text preview: R , then inf E is an element of E . Solution . No, e.g., inf(0 , 1) = 0. (d) Let U n be a sequence of open sets in R and let U = ∞ i n =1 U n . Then every point of U is an interior point of U . Solution . No, e.g., U n = (-1 /n, 1 /n ). Then U = { } , and 0 is not an interior point of U . (e) Any bounded sequence in R converges. Solution . No, e.g., a n = (-1) n . (f) Any Cauchy sequence in a metric space converges. Solution . No, e.g., a n consists of the ±rst n digits of the decimal expansion for √ 2. 1 3. (20 points) Show using the defnition oF convergence that lim n →∞ n 2 n 2 + 1 = 1 . No points will be awarded if the deFnition isn’t used! Solution . Let ǫ > 0. Let N be an integer satisfying N > r (1 /ǫ )-1. Then for n ≥ N , we have v v v v n 2 n 2 + 1-1 v v v v = 1 n 2 + 1 < 1 ( r (1 /ǫ )-1) 2 + 1 = ǫ. This gives the desired convergence. 2...
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This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.

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Fall 2010 Midterm Solutions - R then inf E is an element of...

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