1873_solutions

Log x log x for x sufficiently large we have 1 1 exp

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Unformatted text preview: e inequality fn x f x |  | N. The first of these conditions asserts that the sequence f n converges uniformly to the function f while the second one asserts that f n converges pointwise to f. Make sure that you can distinguish between the two conditions and see that they are not saying the same thing. This exercise departs from the conventional mould. It doesn’t have a solution in the conventional sense. Instead, it asks the student to think about the statements and make sure that the distinction between them has been appreciated. 11. Given that a sequence f n converges uniformly to a function f on a set S and that the function f is bounded, prove that (if we start the sequence at a sufficiently large value of n) the sequence f n converges boundedly to f. Suppose that the sequence f n converges uniformly on a set S to a bounded function f. Choose a number p such that the inequality |f x | p holds for all x S. Using the fact that f n converges uniformly to f, choose N such that the inequality sup|f n f |  1 holds whenever n N. Then for all n N we have sup|f n |  sup|f n f  f | Thus, if we start the sequence f n sup|f n f |  sup|f | p  1. at the integer N then f n converges boundedly to its limit f. 12. Prove that if sequences f n and g n converge uniformly on a set S to functions f and g respectively then 363 f n  g n f  g uniformly on S. Suppose that f n and g n converge uniformly on S to functions f and g respectively. To show that f n  g n f  g uniformly on S, suppose that  0. Choose integers N 1 and N 2 such that sup|f n f |  2 whenever n N 1 and sup|g n g |  2 whenever n N 2 . We define N to be the larger of N 1 and N 2 and we observe that whenever n N we have f  g | sup|f n f |  sup|g n g |   . sup| f n  g n 2 2 13. Give an example of sequences f n and g n that converge uniformly on a set S to functions f and g respectively such that the sequence f n g n fails to converge uniformly to the function fg. Solution: We define f x  x whenever x  0 and for each positive integer n we define fn x  x 1 n whenever x  0. Since sup|f fn |  1 n 0 and 2nx 1 x0 Ý n2 converges uniformly to f on 0, Ý but that f 2 fails to converge uniformly to f 2 on 0, Ý . n sup|f 2 we see that f n f 2 |  sup n 14. Prove that if sequences f n and g n converge uniformly and boundedly on a set S to functions f and g respectively then f n g n fg uniformly on S. Suppose that f n f and g n g uniformly and boundedly on a set S. Choose a number p such that sup|f n | p and sup|g n | p for every n. We observe that sup|f n g n fg |  sup|f n g n fg n  fg n fg | sup|f n g n sup|f n and the latter expression approaches 0 as n sup p|f n Ý. fg n |  sup|fg n fg | f ||g n |  sup|f ||g n f |  sup p|g n g| g| 15. Given that f n is a decreasing sequence of nonnegative continuous functions on a closed bounded set S and that f n converges pointwise to the function 0, prove that f n converges uniformly to the function 0. Solution: We need to show that sup f n 0 as...
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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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