hw29solutions - -M,M show that the sequence g ◦ f n...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: April 26, 2010 Assignment 29 1. Prove that every uniformly convergent sequence of bounded functions is uniformly bounded. That is, { f n } is a uniformly convergent sequence of functions on a set E , then there is a M R so that | f n ( x ) | ≤ M for all x E and all n = 1 , 2 , 3 , . . . . As was stated in class, f n converges to f uniformly if and only if sup | f n - f | = M n converges to 0. Since { M n } converges, this sequence is bounded. Now let L, M be numbers such that M n M and | f 1 | ≤ L for all x . Then, for any x , | f n | ≤ | f n - f 1 | + | f 1 | ≤ | f n - f | + | f - f 1 | + | f 1 | ≤ M n + M 1 + L 2 M + L. Thus, { f n } is uniformly bounded by 2 M + L . 2. Let { f n } be a sequence of functions that converges uniformly to f on A and that satisfies | f n ( x ) | ≤ M for all n N and all x A . If
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Unformatted text preview: -M,M ], show that the sequence { g ◦ f n } converges uniformly to g ◦ f on A . Fix ε > 0. If g is continuous on the compact interval [-M,M ], it is actually uniformly continuous. Thus, there is an η > 0 so that if x,y ∈ [-M,M ] with | x-y | < η , then | g ( x )-g ( y ) | < ε . Moreover, since f n → f uniformly, there is an N ∈ N so that | f n ( x )-f ( x ) | < η for all x ∈ A provided n ≥ N . Combining these, we have that for x ∈ A and n ≥ N , then | g ( f n ( x ))-g ( f ( x )) | < ε which shows the desired uniform convergence. 1...
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