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Unformatted text preview: must be an upper bound of A,
suppose that x is an upper bound of A
a . Using the fact that a is not the largest member of A, choose a
member b of A such that a b. Since x b we see at once that x a and so x is an upper bound of the set
A.
Note that the assumption that A has no largest member isn’t really needed here. All we need to
know is that the specific number a is not the largest member of A. 2. a have exactly the b. Give an example of a set A that has a largest member a such that the sets A and A
exactly the same upper bounds.
We can look at the set 0, 1 Þ 3 .
3. a. Give an example of a set A that has a largest member a such that the sets A and A
same upper bounds.
We can look at the interval 0, 1 . a do not have a. Given that S is a subset of a given interval a, b explain why, for every member x of the set S we have
x  a  b a . Solution: Suppose that x S. We see that
x  x Now since a x a a a  a . x b we see that
x a x a b a and so
x  a  b a . b. Given that a set S of numbers is bounded and that
T x  x S ,
prove that the set T must also be bounded.
Choose a lower bound a of S and an upper bound b of S and then observe from part a that the
number a  b a  is an upper bound of T.
4. Given that A is a set of real numbers and that sup A A, explain why sup A must be the largest member of A. Solution: This assertion is obvious. We are given that inf A is a member of A and we know that
no member of A can be less than inf A, and so inf A is the least member of A.
5. Given that A is a set of real numbers and that inf A A, explain why inf A must be the smallest member of A.
This assertion is obvious. We are given that sup A is a member of A and we know that no member of A can
be larger than sup A, and so sup A is the largest member of A.
6. Is it possible for a set of numbers to have a supremum even though it has no largest member? Solution: You bet it’s possible! That’s the whole point of this chapter!
7. Given that is an upper bound of a set A and that A, explain why sup A.
We are given that is an upper bound of A. Now if w is any number less than then, because
A, the number w can’t be an upper bound of A. Therefore must be the least upper bound of
A.
8. Explain why the empty set does not have a supremum.
Since every number is an upper bound of the empty set, there can’t be a least upper bound of the
empty set.
9. Explain why the set 1, Ý does not have a supremum.
The interval 1, Ý is unbounded above. This set doesn’t have any upper bounds and so it can’t 78 have a least one.
10. Given that two sets A and B are bounded above, explain why their union A Þ B is bounded above.
Using the fact that A is bounded above, choose an upper bound u of A. Now, using the fact that B is
bounded above, choose an upper bound v of B. We now define w to be the larger of the two
numbers u and v. Given any member x of the set A Þ B, there are two possibilities: Either x A, in
which case x u...
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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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