This preview shows page 1. Sign up to view the full content.
Unformatted text preview: rval
x , x contains only finitely many members of the set A Þ B. Therefore no number that lies outside
the set L A Þ L B can be a limit point of A Þ B and we conclude that
L AÞB L A ÞL B .
7. Is it true that if A and B are sets of real numbers then
LA B LA
LB?
What if A and B are closed? What if A and B are open? What if A and B are intervals?
The answers are no, no, no and no. Look at the following example:
A 0, 1 and B 1, 2
These two sets are closed and
LA B L 1
while
LA
L B 0, 1
1, 2 1 .
Now look at the following example:
A 0, 1
and
B 1, 2 .
In this case
LA B L
and
LA
L B 0, 1
1, 2 1 .
8. Is it true that if D R then L D R? Hint: Yes. Suppose that D R. We know that whenever a and b are real numbers and a b there must be 105 members of D lying between a and b. Now suppose that x is a real number. To show that x is a limit
point of D, suppose that 0. Since there must be members of D in the interval x, x we
conclude that the set x , x
D
x is nonempty.
9. Given that a set S of real numbers is nonempty and bounded above but that S does not have a largest member,
prove that sup S must be a limit point of S. State and prove a similar result about inf S.
To show that sup S is a limit point of S, suppose that 0. Since sup S sup S and since sup S is
the least upper bound of S the number sup S fails to be an upper bound of S. Choose a member
x of S such that sup S x. Since x sup S and since sup S does not belong to S we have x sup S.
We conclude that
sup S , sup S
S
sup S
.
10. Given any set S of real numbers, prove that the set L S must be closed. Solution: We shall show that any number that fails to belong to L S must fail to belong to L S .
Suppose that x R L S . Choose a number 0 such that the interval x , x contains only
finitely many members of S. Given any number t in the interval x , x , it follows from the fact that
x , x is a neighborhood of t and the fact that x , x contains only finitely many members of
S that t is not a limit point of S. Thus
x , x
LS
and we have shown, as promised, that x does not belong to L S . 11. Prove that if a set U is open then L U U.
Of course L U
U. Now suppose that x U. To show that x L U , suppose that 0. Using
the fact that x U, choose a number y in the set U
x , x . Using the fact that the set
U
x , x is open, choose 0 such that
y ,y
U
x , x .
We have now found more than one member of U that belongs to the interval x , x and so
we know that
U
x , x
x
.
12. Suppose that S is a set of real numbers, that L S
numbers x and y in S such that x y  . , and that 0. Prove that there exist two different Solution: Choose a limit point t of the set S. Using the fact that the interval t /2, t /2 contains
infinitely many members of S, choose two different members x and y of S that lie in the interval
t /2, t /2 . We observe that x y  . Alternative 6: The Topology of Metri...
View
Full
Document
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

Click to edit the document details