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Unformatted text preview: define x n n and y n 1/n 2 for each n.
b. x n y n 6
We define x n n and y n 6/n for each n.
c. x n y n Ý
We define x n n 2 and y n 1/n for each n.
d. The sequence x n y n is bounded but has no limit.
We define x n n and y n 1 n /n for each n.
7. Given two sequences x n and y n of real numbers such that both of the sequences x n and x n y n are
convergent, is it true that the sequence y n must be convergent?
Yes. Since
yn xn yn
xn
for each n it follows at once that
lim x
lim
lim y n Ý x n y n
nÝ n
nÝ n
Perhaps this exercise should have been given in R k .
8. Given that x n is a sequence of real numbers and that x n 0, prove that
x1 x2 x3 xn
0.
n Solution: Suppose that 0. Using the fact that x n 0 as n Ý, choose an integer N 1 such that
the inequality x n  /2 holds whenever n N 1 . Whenever n N 1 we see that
x 1 x 2 x 3 x n x 1 x 2 x 3 x N 1 x N 1 x N 1 1 x n
n
n
n
x N 1 x N 1 1 x n
x 1 x 2 x 3 x N1
.
n
n
Now we choose an integer N 2 such that 165 N2 2x 1 x 2 x 3 x N 1  and we define N to be the larger of the two numbers N 1 and N 2 . Then whenever n N we have
x N 1 x N 1 1 x n
x 1 x 2 x 3 x N1
x1 x2 x3 xn 0
n
n
n
x 1 x 2 x 3 x N1
x N 1 1  x N 1 2  x n 
n
N2
n N1
.
n
2
2
9. Given that x n is a sequence of real numbers, that x is a real number and that x n
x1 x2 x3 xn
x.
n Solution: Since x n
as n as n x 0 as n Ý we have
x1 x x2 x x3
n Ý. Therefore
x1 x2 x3 xn
n
Ý. x x1 x xn x x2 x x3
n x x, prove that 0 x xn x 0 10. Given that x n and y n are sequences of real numbers and that x n y n 0, prove that x n and y n have
the same set of partial limits.
Since x n x n y n y n and y n x n x n y n for each n the present result follows at once from
Exercise 1.
11. Suppose that x n and y n are sequences of real numbers, that x n y n
a partial limit of at least one of the sequences x n and y n . Prove that
xn
1
yn
as n 0 and that the number 0 fails to be Ý. Solution: From Exercise 10 we know that x n and y n have the same sets of partial limits and we
know, therefore, that 0 is not a partial limit of either of these two sequences.
Using the fact that 0 is not a partial limit of y n , choose an integer N 1 and a number 0 such that the
inequality
y n 
holds whenever n N 1 . For every n N 1 we see that
x n y n 
xn 1 xn yn
yn
yn
and so the fact that
xn
1
yn
as n Ý follows from the sandwich theorem. 12. Give an example to show that the requirement in Exercise 11 that 0 not be a partial limit of at least one of the
two sequences is really needed.
We define x n 2/n and y n 1/n for each n and observe that, even though x n y n 0 as n Ý,
x n /y n 2 as n Ý.
13. Suppose that x n and y n are sequences of real numbers, that x n /y n 1 and that at least one of the
sequences x n and y n is bounded. Prove that x n y n 0. Give an example to...
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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.
 Fall '08
 STAFF
 Math, Calculus

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