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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 oneone 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 oneone 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...
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 Fall '08
 STAFF
 Math, Calculus

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