hw5 - IE 500 Fall 2009 E. Alper Yldrm HOMEWORK 5 (due...

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IE 500 – Fall 2009 E. Alper Yıldırım HOMEWORK 5 (due Wednesday, December 9 in class) 1. Let ( x n ) be a sequence of real numbers. Let -∞ < lim inf n →∞ x n = a < lim sup n →∞ x n = b < + . (a) For each real number ± > 0, prove that there exist infinitely many elements of the sequence ( x n ) such that x n ( a - ±,a + ± ). (b) For each real number ± > 0, prove that there exist infinitely many elements of the sequence ( x n ) such that x n ( b - ±,b + ± ). 2. Using the result from the first problem, prove that there exist two subsequences ( a n ) and ( b n ) of the sequence ( x n ) such that lim n →∞ a n = a and lim n →∞ b n = b . 3. Let ( x n ) be a sequence of real numbers such that -∞ < lim inf n →∞ x n = a lim sup n →∞ x n = b <
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This note was uploaded on 04/15/2010 for the course INDUSTRIAL ie513 taught by Professor Zeynephuygur during the Spring '10 term at Bilkent University.

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