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Unformatted text preview: e fact that x is an interior point of U, choose 0 such that
x , x
U.
Since U V we therefore have
x , x
V.
Therefore x is an interior point of V.
4. Suppose that x is a real number and that U R. Prove that the following two conditions are equivalent: a. The set U is a neighborhood of the number x.
b. It is possible to find two numbers a and b such that
x
a, b U. Solution: To prove that condition a implies condition b we assume that U is a neighborhood of the
number x. Choose 0 such that
By defining a x x , x
U.
and b x we obtain two numbers a and b such that a b and such that
x
a, b
U. Now to prove that condition b implies condition a we assume that condition b holds. Choose numbers a
and b such that a b and such that
x
a, b
U.
Using the fact that the interval a, b is a neighborhood of x, choose 0 such that
x , x
a, b .
Since x , x
U we have shown that U is a neighborhood of x.
5. Suppose that x and y are two different real numbers. Prove that it is possible to find a neighborhood U of x
and a neighborhood V of y such that U V . Solution: We may assume, without loss of generality, that x y. Choose a number c between x and
y. The intervals Ý, c and c, Ý are, respectively, neighborhoods of x and y and the intersection of these
two intervals is empty.
6. Given that S is a set of real numbers and that x is an upper bound of S, explain why S cannot be a
neighborhood of x. Solution: If is any positive number then, since all of the numbers between x and x must lie in
R S, the interval x , x cannot be included in S. 7. Given that a set S of real numbers is nonempty and bounded above, explain why neither S nor R
neighborhood of sup S. S can be a Solution: We see from Exercise 6 that S is not a neighborhood of sup S. Now we observe that,
96 whenever 0, since the number x fails to be an upper bound of S, there must be members of S in the
interval x , x . Therefore, whenever 0, the interval x , x fails to be included in the set R S
and so R S must also fail to be a neighborhood of x.
8. Suppose that A and B are sets of real numbers and that x is an interior point of the set A Þ B. Is it true that x
must either be an interior point of A or an interior point of B? Hint: The answer is no. Give an example to show what can go wrong.
9. Suppose that A and B are sets of real numbers and that x is an interior point both of A and of B. Is it true that x
must be an interior point of the set A B?
Yes it is true. Suppose that x is an interior point of both of the sets A and B. Choose a number
1 0 such that
x 1, x 1
A
and choose a number 2 0 such that
x 2, x 2
B.
We now define to be the smaller of the two numbers 1 and 2 and we observe that
x , x
A B.
10. Suppose that x and y are real numbers and that U is a neighborhood of y. Prove that the set V defined by
V xu u U
is a neighborhood of the number x y. Solution: We need to find a number 0 such that the i...
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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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