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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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 Fall '08
 STAFF
 Math, Calculus

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