Unformatted text preview: it follows from the facts that x x sup A inf B y
which tells us that x y. Therefore it is impossible to find x A and y
have reached the desired contradiction. sup A and inf B y B such that x y and we 8. Suppose that A is a nonempty bounded set of real numbers, that A has no largest member and that x sup A.
Prove that there are at least two different members of A lying between x and sup A.
Since x is less than the least upper bound of A we know that x can’t be an upper bound of A. Using
the fact that x isn’t an upper bound of A we choose a member a of A such that x a. Since A has
no largest member it must have a member larger than the member a. We choose a member b of A
such that a b. Since A has members greater than b we deduce that
x a b sup A
and we have found two different members of A between x and sup A.
9. Suppose that A is a nonempty bounded set of real numbers, that 0 and that for any two different members
x and y of A we have |x y | . Prove that A has a largest member. You can find a hint to the solution of
this exercise in a forthcoming theorem.
10. Suppose that S is a nonempty bounded set of real numbers, that inf S and sup S, and that every
number that lies between two members of S must also belong to S. Prove that S must be one of the four
intervals , , , , , , , . See a coming theorem for a solution of this exercise.
11. Suppose that A is a nonempty bounded set of real numbers, that inf A and that sup A. Suppose that
S x y x A and y A .
Prove that sup S . You will find a solution to this exercise in the next section. 12. Suppose that A is a set of numbers and that A is nonempty and bounded above. Suppose that q is a given
number and that the set C is defined as follows:
C qx x A .
Prove that the set C is nonempty and bounded above and that
sup C q sup A
This exercise is a special case of the next exercise because C A q .
13. Suppose that A and B are nonempty bounded sets of numbers and that the sets A B and A
above. Prove that
sup A B sup A sup B
sup A B sup A inf B B are defined as Solution: We shall prove that
sup A B sup A sup B
Step 1: We want to show that the number sup A sup B is an upper bound of the set A B. Suppose that 81 x A B. Using the definition of A B we choose a number a A and a number b
x a b. Since a sup A and b sup B we see that
x a b sup A sup B. B such that Step 2: We want to show that the number sup A sup B is actually the least upper bound of the set A B.
Suppose that u is any upper bound of the set A B. Given any number a A and b B we have
a b u. Therefore, whenever b B we know that the inequality
holds for all a A. Therefore, whenever b B, the number u b is an upper bound of A and must satisfy
sup A u b
which we can also write as
b u sup A.
We conclude that u sup A is an upper bound of B and so
sup B u sup A
which we can write as
sup B sup A u.
Thus sup A sup...
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