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204PS22008

204PS22008 - Econ 204 Summer/Fall 2008 Problem Set 2 Due in...

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Unformatted text preview: Econ 204, Summer/Fall 2008 Problem Set 2 Due in Lecture Tuesday, August 5 1. Cluster Points Prove that the set of cluster points of a sequence { x k } is closed. 2. Boundary/Interior/Closure Find the boundary, interior, and closure of the following sets: (a) { ( x, y ) ∈ R 2 | x 2 − y 2 > 5 } . (b) { ( x, y ) ∈ R 2 | x ≥ y } . (c) { ( x, y ) ∈ R 2 | x ∈ Q } , where Q denotes the rational numbers. 3. Convergence of Sequences Consider the sequence defined by a = 1 , a 1 = 1 + 1 1+ a , . . . , a n = 1 + 1 1+ a n- 1 , . . . . Show that this sequence converges to √ 2. (HINT: Show first that the subsequences a , a 2 , a 4 , . . . and a 1 , a 3 , a 5 , . . . each converge, and then show that the limit of each subsequence is equal to √ 2.) 4. Normed Spaces Consider l ∞ , the set of bounded sequences (so a ∈ l ∞ if and only if a = ( a 1 , a 2 , . . . ) with a i ∈ R for all i and there exists some M < ∞ such that | a i | ≤ M for all i )....
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204PS22008 - Econ 204 Summer/Fall 2008 Problem Set 2 Due in...

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