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Unformatted text preview: ers and that a b, that H is a closed subset of R and that x
a, b
Prove that there exist numbers u and v such that a u x v b and u v and u, v
H . a u x v H. b Using the fact that x belongs to the open set R H, choose 0 such that
x , x
H .
We now define u to be the larger of the numbers a and x /2 and v to be the smaller of the
numbers b and x /2.
2. Suppose that H n is a sequence of closed subsets of an interval a, b where a b and that none of
the sets H n has any interior points. Find a contracting sequence of subintervals a n , b n of the interval a, b
such that
Hn
an, bn
for each n. By looking at a number that lies in the intersection of all the intervals a n , b n , prove that there
exists a number in the interval a, b that does not belong to the set
Ý Hn.
n1 a, b , choose a number x 1
a, b
H 1 . Using Exercise 1, choose
Using the fact that the set H 1
two numbers a 1 and b 1 such that a 1 b 1 and a a 1 x 1 b 1 b and
H1
a1, b1 .
Since a 1 b 1 and since the set H 2 has no interior point, the open interval a 1 , b 1 must contain
numbers that do not belong to H 2 . Choose x 2
H 2 . We now use Exercise 1 again to
a1, b1
choose two numbers a 2 and b 2 such that a 2 b 2 and a 1 a 2 x 2 b 2 b 1 . Continuing in this
way we obtain the desired sequence of intervals a n , b n .
3. Suppose that H n is a sequence of closed subsets of an interval a, b where a b and that none of the sets
H n has any interior points. Prove that the set
Ý Hn a, b n1 is dense in the interval a, b . Hint: If c, d is a subinterval of the interval a, b , apply Exercise 2 to the sequence of sets c, d .
Hn
We can observe that whenever c, d is a nonempty open interval of a, b , the set
Ý c, d Hn . n1 4. Suppose that H n is a sequence of closed sets, that a b and that 366 a, b Ý Hn.
n1 For each n, suppose that U n is the set of interior points of the set H n . Prove that the set
Ý Un
n1 is a dense open subset of the interval a, b .
Suppose that c, d is a nonempty open subinterval of a, b . Since
n Hn c, d c, d , n1 it follows from Exercise 2 that at least one of the set H n
interior point of H n
c, d must belong to U n . c, d must have an interior point and an 5. Suppose that f n is a sequence of continuous functions on an interval a, b where a b and that f n
converges pointwise on a, b to a function f. Suppose that 0 and that for each n, suppose that H n is
defined to be the set of all those numbers x
a, b for which the inequality
f i x f j x 
3
holds whenever i n and j n. Suppose finally that U n is the set of interior points of H n for each n.
a. Prove that the set
V Ý Un
n1 is an open dense subset of the interval a, b .
This assertion follows at once from Exercise 4 and the fact that
a, b Ý Hn.
n1 b. Prove that for every number x V there exists a number 0 such that for every number t satisfying
the inequality t x  we have f t f x 
.
Choose n such that x U n . Choose 1 0 such that x , x...
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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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