1873_solutions

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online