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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