hw4 - IE 500 Fall 2009 E. Alper Yldrm HOMEWORK 4 (due...

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IE 500 – Fall 2009 E. Alper Yıldırım HOMEWORK 4 (due Wednesday, November 18 in class) 1. Give a complete characterization of Cauchy sequences of integers . Discuss what kind of real numbers can be represented by such Cauchy sequences. 2. Let ( x n ) be a Cauchy sequence of real numbers. Consider the seqence ( y n ) given by y 1 = 0 and y n = x n - x n - 1 , n = 2 , 3 ,... . (a) Prove that ( y n ) is a Cauchy sequence of real numbers. (b) Find the limit of the sequence ( y n ) and justify your result. 3. (a) Let a and b be any two real numbers. Prove that || a | - | b || ≤ | a - b | . (b) Let ( x n ) be a Cauchy sequence of real numbers. Using part (a), prove that (
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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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