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Unformatted text preview: y is in the set S and it is clear that
xn
Ý.
2. Suppose that S is a nonempty set of real numbers and that is an upper bound of S. Prove that the following
conditions are equivalent:
a. We have sup S.
b. There exists a sequence x n in S such that x n as n Ý.
In Exercise 7 of the exercises on closure we saw that an upper bound of set S is close to S
if and only if is equal to sup S. We also saw in a recent theorem that a number is close to a
set S if and only if there exists a sequence in S that converges to . The present exercise
follows at once from these two facts. Of course, we can write this exercise out directly if we
wish.
3. Given a sequence x n that is frequently in a set S of real numbers, and given a partial limit x of the sequence
x n , is it necessarily true that x S?
No, this statement need not be true. If we define x n 1 n for each n then, although x n is
frequently in the set 1 it has the partial limit 1 that is not close to 1 .
4. Prove that a set U of real numbers is open if and only if every sequence that converges to a member of U must
be eventually in U.
We know from a recent theorem that a set H is closed if and only if no sequence that is frequently
in H can have a limit that doesn’t belong to H. Therefore if U is a set of real numbers then the set
R U is closed if and only if no sequence with a limit in U can fail to be eventually in U.
Of course the exercise can also be done directly.
5. Given that S is a set of real numbers and that x is a real number, prove that the following conditions are
equivalent:
a. The number x is a limit point of the set S.
b. There exists a sequence x n in the set S x such that x n x. Solution: The assertion in this exercise follows at once from the corresponding theorem about
limits of sequences and closure of a set, and from the fact that x is a limit point of S if and only if
xS
x. 143 6. Prove that if x n is a sequence of real numbers then the set of all partial limits of x n is closed.
Suppose that x n is a sequence of real numbers and write the set of partial limits of x n as H. In
order to show that H is closed we shall show that R H is open. Suppose that x R H. In other
words, suppose that x is a number that is not a partial limit of x n . Choose a number 0 such
that the condition
xn
x ,x
holds for at most finitely many integers n. Given any number y
x , x , it follows from the fact
that x , x is a neighborhood of y and the fact that x n belongs to this neighborhood of y for at
most finitely many integers n that y is not a partial limit of x n . In other words,
x ,x
RH
and we have shown, as promised, that the set R H is open.
7. Suppose that A and B are nonempty sets of real numbers and that for every number x
y B we have x y. Prove that the following conditions are equivalent: A and every number a. We have sup A inf B.
b. There exists a sequence x n in the set A and a sequence y n in the set B such that y n
n Ý. xn 0 as Solution: From the information gi...
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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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