1873_solutions

Fact that 2 lim uu 0 e converges pointwise to the

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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.

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