ecg765pset2

ecg765pset2 - sequence is any infinite subset of the...

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Atsushi Inoue Econ 765: Mathematical Methods for Economists Fall 2009 Problem Set ] 2 Due in class on Tuesday, September 8, at 11:45am When you answer each of the following questions, please use formal ar- guments (e.g., use definitions and theorems rather than pictures only). 1. Define { x n } n =1 in < 2 by x n = " 1 n cos( n ) 1 n sin( n ) # . Find if the sequence is convergent. If it is convergent, find its limit. (2 points) 2. Let a sequence { x n } n =1 in < k be given. Let m be any rule that assigns to each n ∈ N a value m ( n ) ∈ N . Suppose further that m is strictly increasing, i.e., for each n ∈ N , we have m ( n ) < m ( n + 1). Given { x n } we can now define a new sequence { x m ( n ) } . This new sequence is called a subsequence of { x n } . Put it differently, a subsequence of a
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Unformatted text preview: sequence is any infinite subset of the original sequence that preserves the ordering of terms. Give an example of a nonconvergent sequence { x n } such that { x n } con-tains at least one convergent subsequence and every convergent subse-quence of { x n } converges to the limit x = 0. ( Hint: You need to prove that (i) your example is nonconvergent, that (ii) it contains at least one convergent subsequence and that (iii) every convergent subsequence of it converges to 0.) (6 points) 3. Do exercise 12.9 of Simon and Blume (1994, p.260). (2 points) 1...
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