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Unformatted text preview: ed above we know that that A
is bounded above. Finally, since sup B, being an upper bound of B, must be an upper bound of A,
and since sup A is the least upper bound of A we have sup A sup B.
3. Given that A is a nonempty bounded set of numbers, explain why inf A sup A.
Using the fact that A is nonempty we choose a member a of A. We know that
inf A a sup A.
4. It is given that A and B are nonempty bounded sets of real numbers, that for every x A there exists y
such that x y and for every y B there exists x A such that y x. Prove that sup A sup B. B Solution: We need to show that the two sets A and B have exactly the same upper bounds and for this
purpose we shall show that a number fails to be an upper bound of A if and only if it fails to be an upper
bound of B.
Suppose that u fails to be an upper bound of A. Choose a member x of A such that u x. Using the given
property of A and B we now choose a member y of the set B such that x y and, since u x y, we
conclude that u can’t be an upper bound of B. We can show similarly that a number that fails to be an
upper bound of B must also fail to be an upper bound of A.
5. Suppose that A and B are nonempty sets of real numbers and that for every x A and every y B we have
x y. Prove that sup A inf B. Give an example of sets A and B satisfying these conditions for which
sup A inf B.
Given any member y of the set B it follows from the fact that x y for every x A that y must be an
upper bound of A. In other words, every member of B is an upper bound of A and the fact that B is
nonempty shows that A is bounded above. A similar argument shows that every member of A is a
lower bound of B and the fact that A is nonempty guarantees that B is bounded below. Thus sup A
and inf B exist.
Given any member y of B, the fact that y is an upper bound of A and sup A is the least upper bound
of A tells us that sup A y. In other words, sup A is a lower bound of B. Since inf B is the greatest
lower bound of B we deduce that sup A inf B.
6. Suppose that A and B are nonempty sets of real numbers and that sup A inf B. Prove that for every number
0 it is possible to find a member x of A and a member y of B such that x y. Solution: Suppose that 0. Using the fact that
sup A inf B inf B
and that sup A is therefore not a lower bound of B, choose a member y of the set B such that
sup A y.
From the fact that
y sup A
we deduce that y is not an upper bound of A and, using this fact, we choose a member x of A such that
y x. In this way we have found x A and Y B such that x y. 80 inf B
y−δ x sup A sup A + δ y 7. Suppose that A and B are nonempty sets of real numbers, that sup A inf B and that for every number 0 it
is possible to find a member x of A and a member y of B such that x y. Prove that sup A inf B. Solution: To obtain a contradiction, assume that sup A inf B and define
We observe that 0. Now for all x
that inf B sup A.
A and y B,...
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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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