1873_solutions

Positive integer then 5n 5 n 2 3n 1 5n 2 1 2 4n 4n

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Unformatted text preview: er and that for every positive integer n we have y n  x n p . Prove that y n x as n Ý. Suppose that  0. Using the fact that x n x as n Ý we choose an integer N such that whenever n N we have |x n x |  . We observe that the inequality |y n x |  will hold whenever 138 np N which is the same as saying that n N p. 9. Given that a n b n for every positive integer n and given that a n Ý, prove that b n Ý. Suppose that w is a real number. Using the fact that a n Ý as n Ý we choose an integer N such that the inequality a n  w holds whenever n N. Then whenever n N we have b n a n  w. 10. Suppose that a n and b n are sequences of real numbers and that |a n b n | 1 for every positive integer n and that Ý is a partial limit of the sequence a n . Prove that Ý is a partial limit of b n . We know that b n a n 1 for each n. Suppose that w is a real number. Since Ý is a partial limit of a n there are infinitely many integers n for which a n  w  1 and for all such integers we have b n  w. 11. Two sequences a n and b n of real numbers X are said to be eventually close if for every number there exists an integer N such that the inequality d a n , b n  holds for all integers n N. 0 a. Prove that if two sequences a n and b n are eventually close and if a number x is the limit of the sequence a n then x is also the limit of the sequence b n . Suppose that a n and b n are eventually close and that a n x as n Ý. To show that b n x as n Ý, suppose that  0. Choose N 1 such that the inequality |a n x |  2 holds whenever n N 1 . Choose N 2 such that the inequality |a n b n |  2 holds whenever n N 2 . We define N to be the larger of N 1 and N 2 and we see that whenever n N we have  . |b n x | |b n a n |  |a n x |  2 2 b. Prove that if two sequences a n and b n are eventually close and if a number x is a partial limit of the sequence a n then x is also a partial limit of the sequence b n . Suppose that a n and b n are eventually close and that x is a partial limit of a n . Suppose that  0. Choose an integer N such that the inequality |a n b n |  2 holds whenever n N. Since there are infinitely many integers n for which |a n x |  2 there must be infinitely many integers n N for which |a n x |  2 . For every one of these integers n we have |b n x| |b n a n |  |a n x|  2  2 . 12. Suppose that a n and b n are sequences of real numbers, that a n a and b n b as n Ý, and that a  b. Prove that there exists an integer N such that the inequality a n  b n holds for all integers n N. Choose a number p between a and b. a p b Using the fact that a n a as n Ý and the fact that the interval Ý, p is a neighborhood of a, choose an integer N 1 such that a n  p whenever n N 1 . Similarly, choose an integer N 2 such that b n  p whenever n N 2 . We define N to be the larger of N 1 and N 2 and we observe that a n  p  b n whenever n N. 13. Give an example of two sequences a n and b n of real numbers satisfying a n  b n for every positive integer n even...
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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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