f07HW4 - f uniformly on S and g n g uniformly on S . Then,...

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MAT 125A Homework 4 M.Fukuda Please submit your answers at the discussion session on November 1st Thursday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. Let f n ( x ) = p x - 1 n P 2 for x [0 , 1]. 1) Show that the sequence of functions { f n } converges to some function pointwise on [0 , 1] and write the limit function. 2) Does { f n } converges to f uniformly on [0 , 1]? Prove your idea. Let g n ( x ) = x n 1 + x n for x [0 , + ). 3) Repeat 1) for g on x [0 , + ). 4) Repeat 2) for g on x [0 , + ). 2. Let S R and f, f n , g, g n : S R . Suppose f n
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Unformatted text preview: f uniformly on S and g n g uniformly on S . Then, 1) Show that, for any real number c , cf n cf uniformly on S . 2) Show that f n + g n f + g uniformly on S . Let S = R , f n ( x ) = x , f ( x ) = x , g n ( x ) = 1 /n and g ( x ) = 0. 3) Show that f n g n does not converge to fg uniformly on S. 3. Let S R and f, f n : S R . Suppose f n are all uniformly continuous on S and f n f uniformly on S . Show that f is uniformly continuous on S . Hint: See the proof of Thm 24.3. 1...
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