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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online