Unformatted text preview: 2 3
holds when 2/n .
Using the fact that the number 2/ is not an upper bound of the set Z of integers we choose an
integer N such that N 2/ , in other words,
Then, whenever n N we have
3 2 3
3 3 2
and so we have shown that x n is eventually in 3 , 3 .
3. Given that x n 1/n for each positive integer n and that x 0, prove that x is not a partial limit of x n . Solution: In the event that x 0, the interval Ý, 0 is a neighborhood of x and it is clear that x n
fails to be frequently in this neighborhood. Therefore no negative number can be a partial limit of x n .
Suppose now that x 0. The interval x/2, Ý is a neighborhood of x x
2 0 x and the condition x n
x/2, Ý must fail to hold whenever n 2/x. Therefore the sequence x n cannot be
frequently in the interval x/2, Ý and the number x cannot be a partial limit of x n .
4. Given that
1 n n 3 if n is a multiple of 3
xn 0 if n is one more than a multiple of 3 , 4 if n is two more than a multiple of 3 Prove that the partial limits of x n are Ý, Ý, 0 and 4.
Since x n is unbounded both above and below, it follows from the discussion of infinite partial
limits we saw earlier that both Ý and Ý are partial limits of x n . Since the equation x n 0 holds
for infinitely many values of n we know that x n is frequently in every neighborhood of 0 and so 0 is
a partial limit of x n . In the same way we can see that 4 is a partial limit of x n .
Now we need to explain why any real number other than 0 and 4 must fail to be a partial limit of
x n . Suppose that x R
0, 4 .
In the event that 0 x 4, the fact that x is not a partial limit of x n follows from the fact that x n is
not frequently (or ever) in the interval 0, 4 which is a neighborhood of x.
Now suppose that x 0. x 1 x 0 In order to show that x is not a partial limit of x n we shall make the observation that x n is not
frequently in the interval x 1, 0 which is a neighborhood of x. In fact, the inequality
x 1 xn 0
can hold only when n is odd and
x 1 n3
which is equivalent to saying that n
x. Since there are only finitely many such positive
integers n we conclude that x n is not frequently in the interval x 1, 0 134 Finally we must consider the case x 4. x 4 x1 In this case we observe that there can be only finitely many positive integers n for which
4 n3 x 1
and so, once again, x can’t be a partial limit of x n .
5. Give an example of a sequence of real numbers whose set of partial limits is the set 1 Þ 4, 5 . Hint: For each positive integer n, if n can be written in the form
for some positive integers m and k and if
k 5 then we define
xn m .
In all other cases we define x n 1. Observe that the range of the sequence x n is the set
1 Þ Q 4, 5
and then show that the set of partial limits of x n is 1 Þ 4, 5 .
Since the equation x n 1 holds for infinitely many positive integers n the number 1 must be a
partial limit of x n . To see that every number in the interval 4,...
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