IE 500 – Fall 2009E. Alper YıldırımHOMEWORK 4 (due Wednesday, November 18 in class)1. Give a complete characterization of Cauchy sequences ofintegers. Discuss what kind of real numberscan be represented by such Cauchy sequences.2. Let (xn) be a Cauchy sequence of real numbers.Consider the seqence (yn) given byy1= 0 andyn=xn-xn-1,n= 2,3, . . ..(a) Prove that (yn) is a Cauchy sequence of real numbers.(b) Find the limit of the sequence (yn) and justify your result.3.(a) Letaandbbe any two real numbers. Prove that||a| - |b|| ≤ |a-b|.(b) Let (xn) be a Cauchy sequence of real numbers. Using part (a), prove that (
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