1873_solutions

# Range being finite has no limit point c prove that if

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Unformatted text preview: 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 in a metric space 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 in a metric space X are eventually close and if a point 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 d a n , x  2 holds whenever n N 1 . Choose N 2 such that the inequality d 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 d bn, an  d an, x   . d bn, x 2 2 b. Prove that if two sequences a n and b n in a metric space X are eventually close and if a point 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 d a n , b n  2 holds whenever n N. Since there are infinitely many integers n for which d a n , x  2 there must be infinitely many integers n N for which d a n , x  2 . For every one of these integers n we have d bn, x d bn, an  d an, x  2  2 . 12. Suppose that a n and b n are sequences of real numbers, that a n a and that 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 though the sequence a n has a partial limit a that is less than a partial limit b of the sequence bn . We define a n  2  1 n and b n  3  1 n for every positive integer n. We see that a n  b n for 163 each n and that 3 is a partial limit of a n and that 2 is a partial limit of b n . Some Exercises on the Algebraic Rule...
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