1873_solutions

# Members of the set q 4 5 we know that the condition x

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Unformatted text preview: have d x, y | d x, x n  d y n , y   . |d x n , y n 2 2 14. Prove that if x n is a sequence in R k and x R k and if x n x as n Ý then x n x as n Ý. Since x n  x n O and x  x O the desired result follows at once from Exercise 13. 15. Prove that if a sequence x n in a metric space X converges to a point x X then every subsequence of x n also converges to x. Suppose that x n is a sequence converging to a point x in a metric space X and that x n i is a subsequence of x n . Suppose that  0 and, using the fact that x n x as n Ý, choose an integer N such that the inequality d x n , x  holds whenever n N. Using the fact that n i is a strictly increasing sequence of positive integers, choose an integer k such that the inequality n i N will hold whenever i k. Then, whenever i k we have d x n i , x  . 16. a. Prove that if x n is a sequence in a metric space X and if a point x is a limit point of the range of x n then x must be a partial limit of the sequence x n . Solution: Suppose that x n is a sequence in a metric space X and that a point x is a limit point of the range of x n . Suppose that  0. Since there are infinitely many members of the range of x n that lie in the ball B x, , there must be infinitely many positive integers n for which x n B x, . b. Give an example of a sequence x n in a metric space X and a partial limit x of x n such that x fails to be a limit point of the range of the sequence x n . 160 If a sequence is constant then it converges but its range, being finite, has no limit point. c. Prove that if a sequence x n in a metric space X is one-one and if a point x is a partial limit of the sequence x n then x must be a limit point of the range of x n . Suppose that x n is a sequence in a metric space X and that x is a partial limit of x n . Suppose that  0. Since there are infinitely many integers n for which x n B x, and since the sequence x n is one-one we see that there is more than one member of the range of x n in the ball B x, . Therefore x is a limit point of the range of x n . 17. For each positive integer n, if n can be written in the form n  2m3k where m and k are postive integers and m k then we define xn  m . k Otherwise we define x n  0. Prove that the set of partial limits of the sequence x n is 0, 1 . Since the range of x n is the set of all rational numbers in the interval 0, 1 , every neighborhood of a number x in the interval 0, 1 must contain infinitely many members of the range of x n and must, therefore, contain the number x n for infinitely many integers n. Thus every member of the interval 0, 1 is a partial limit of x n . If x is any number in the set R 0, 1 then, since R 0, 1 is a neighborhood of x and x n is not frequently (or, indeed, ever) in the set R 0, 1 , the number x must fail to be a partial limit of x n . Exercises on the Elementary Properties of Limits 1. The purpose of this exercise is to use Scientific Notebook to gain an intuitive feel for the limit behaviour of a rather difficult seque...
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## This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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