1873_solutions

# Words 2 n then whenever n n we have 156 3 3 2 n is

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ill hold whenever n N. Then, whenever n N we have x ,x  x 5 ,x  5 . xn Now to prove that condition b implies condition a we assume that condition b holds. To prove that condition a holds, suppose that  0. Using the fact that /5 is a positive number, we choose an integer N such that the condition xn holds whenever n ,x  5 5 5 N we have x N. Thus for every n xn x 5 5 5 ,x  5 5 x ,x  . 9. Prove that n 2  3n  1 2n 2  n  4 1 2 as n Ý. We begin by observing that if n is any positive integer then 5n  5 . n 2  3n  1 5n 2 1 2 4n 4n 2 2n 2  n  4 4n 2  2n  8 Now suppose that  0. The inequality n 2  3n  1 1 2 2n 2  n  4 will hold when 5 4n which requires that n 5 4 Choose an integer N such that N  5/ 4 and observe that, whenever n N we have 5 5. n 2  3n  1 1 2 4n 4N 2n 2  n  4 10. Given that x n is a sequence in a metric space X and that x X, prove that x n x as n Ý if and only if d xn, x 0 The condition x n x as n Ý says that for every  0 there exists an integer N such that whenever n N we have d x n , x  . The condition d x n , x 0 as n Ý says that for every  0 there exists an integer N such that whenever n N we have |d x n , x 0 |  . These two conditions clearly say the same thing. as n Ý. 11. Suppose that x n is a sequence in a metric space X, that x X, and that y n is a sequence of real numbers converging to 0. Suppose that the inequality d x, x n yn holds for every n. Prove that x n x as n Ý. Suppose that  0. Using the fact that y n 0 as n Ý, choose an integer N such that the inequality y n  will hold whenever n N. We see at once that the inequality d x n , x  also holds whenever n N. 159 12. Given that x n and y n are sequences of real numbers, that x n y n for each n and that y n Ý as n Ý, prove that x n Ý as n Ý.. Suppose that w is a real number. Using the fact that y n Ý as n Ý we choose an integer N such that the inequality y n  w holds whenever n N. Then whenever n N we have x n y n  w. 13. Suppose that x n and y n are sequences in a metric space X and that x and y are points of X. a. Prove that for every n we have d x, y d x, x n  d x n , y n  d y n , y and deduce that d x, x n  d y n , y . d x, y d x n , y n Then prove that for every n we have d x, y d x, x n  d y n , y d xn, yn and deduce finally that d x, y | d x, x n  d y n , y . |d x n , y n Given any n we have d x, y d x, x n  d x n , y d x, x n  d x n , y n  d y n , y and the inequality d x, x n  d y n , y . d x, y d x n , y n follows at once. In the same way we can show that d x, y d x n , x  d y, y n . d xn, yn Therefore d x, y | d x, x n  d y n , y . |d x n , y n d x, y as n Ý. b. Prove that if x n x and y n y as n Ý then d x n , y n We assume that x n x and y n y as n Ý. Suppose that  0. Choose an integer N 1 such that the inequality d x n , x  2 holds whenever n N 1 . Choose an integer N 2 such that the inequality d y n , y  2 holds whenever n N 2 . We define N to be the larger of the integers N 1 and N 2 . Then whenever n N we...
View Full Document

## 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.

Ask a homework question - tutors are online