hw2solutions

hw2solutions - Math 318 HW #2 Solutions 1. Exercise 6.2.16....

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1. Exercise 6.2.16. For each n N , let f n be a function defined on [0 , 1]. If ( f n ) is bounded on [0 , 1]—that is, there exists an M > 0 such that | f n ( x ) | ≤ M for all n N and x [0 , 1]—and if the collection of functions ( f n ) is equicontinuous (Exercise 6.2.15), follow these steps to show that ( f n ) contains a uniformly convergent subsequence. (a) Use Exercise 6.2.14 to produce a subsequence ( f n k ) that converges at every rational point in [0 , 1]. To simplify the notation, set g k = f n k . It remains to show that ( g k ) converges uniformly on all of [0 , 1]. Proof. Since ( f n ) is bounded and Q [0 , 1] is countable, Exercise 6.2.14 implies that there is a subsequence ( f n k ) that converges at all points in Q [0 , 1]. (b) Let ± > 0. By equicontinuity, there exists a δ > 0 such that g k ( x ) - g k ( y ) | < ± 3 for all | x - y | < δ and k N . Using this δ , let r 1 , r 2 , . . . , r m be a finite collection of rational points with the property that the union of the neighborhoods V δ ( r i ) contains [0 , 1]. Explain why there must exist an N N such that | g s ( r i ) - g t ( r i ) | < ± 3 for all s, t N and r i in the finite subset of [0 , 1] just described. Why does having the set { r 1 , r 2 , . . . , r m } be finite matter? Proof. Let ± > 0. Since r 1 , . . . , r m Q [0 , 1], we know that each sequence ( g k ( r i )) converges and, hence, is Cauchy. Therefore, for each i = 1 , . . . , m , there exists N i N such that r, s N i implies that | g s ( r i ) - g t ( r i ) | < ± 3 . Let N = max { N 1 , . . . , N m } (this is where it’s essential that the collection { r 1 , . . . , r m } is finite; if it were infinite, then there might well not be a maximum N i ). Then, for any r, s N and for any i ∈ { 1 , . . . , m } , we have that s, t N i and so | g s ( r i ) - g t ( r i ) | < ± 3 , as desired. (c) Finish the argument by show that, for an arbitrary x [0 , 1], | g s ( x ) - g t ( x ) | < ± for all s, t N . 1
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hw2solutions - Math 318 HW #2 Solutions 1. Exercise 6.2.16....

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