psAnalysis1Qu - Fall 2007 ARE211 Problem Set #01 First...

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Unformatted text preview: Fall 2007 ARE211 Problem Set #01 First Analysis Problem Set Due date: Sep 11 Problem 1 Please use the Pythagorean metric in the following problem a) Consider the sequence x n = 2 + (- 1) n n defined on R . Prove (i) that the sequence is a convergent sequence using the definition of a convergent sequence and show (ii) that the sequence is a Cauchy sequence using the definition of a Cauchy sequence . b) Now consider the sequence x n = 2 + (- 1) n n defined on S = R \ { 2 } . Using your proof from part a) argue that it is still a Cauchy sequence in S . Prove that it is not a convergent sequence in S . (Note: The set A \ B is defined as: A \ B = { x | x A , x / B } ). Problem 2 a) Prove that a sequence x n in X converges in the discrete metric if and only if there exists x X and a N N such that for all n > N , x n = x . b) In class we showed that every Cauchy sequence in R with respect to the Pythagorean metric is also a convergent sequence in R with respect to the Pythagorean metric . Show that every Cauchy sequence in R with respect to the discrete metric is also a convergent sequence in R with respect to the discrete metric ....
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.

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psAnalysis1Qu - Fall 2007 ARE211 Problem Set #01 First...

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