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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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