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EXERCISES 8

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EXERCISES 8.2 1. Prove Theorem 8.2.5. 2. a. If {fn} and {8n} converge uniformly on a set E, prove that {fn + 8n} converges uniformly on E. *b. If {fn) and (8n} converge uniformly on a set E, and there exist constants M and N such that f(x) = M and 18.(*)| S N for all n E N and all x € E, prove thai {fn8n} converges uniformly on E. c. Find examples of sequences {fr} and {8n} that converge uniformly on a set E. but for which {fn8n} does not converge uniformly on E. 3. Show that if {fr} converges uniformly on (a, b) and {fo(a)} and {fr(6)} converge, then {fr} converges uniformly on (a, b). 4. *Let fa(x) = nx{1 – x?)",0 SXSI. Show that {fn} does not converge uniformly to 0 on (0, 1). r 5. Let f (x) 0sxsl. 1 + x *a. Show that {f} converges uniformly to 0 on (0, a) for any a, 0 < a < 1. b. Does {n} converge uniformly on (0, 1)? 8.2 Uniform Convergence 329 6. Show that the sequence {nxe nx} converges uniformly to 0 on (a. 00) for every a > 0. 7. For each n EN, set X fu(x) = x + sin nr, XER. n Show that the sequence {fr} converges uniformly to f(x) = x for all x € (-a, a), a > 0. Does {{n} converge uni formly to f on R? EXERCISES 8.3 1. *Show that the series XX-ox(1 – x)* cannot converge uniformly for 0 sxsl. 2. Forn E N. let fm(x) = x"/(1 + x"), x € (0, 1). Prove that the sequence {fr} does not converge uniformly on (0, 1). 3. Give an example of a sequence {fr} of functions on (0, 1), such that each fn is not continuous at any point of (0, 1). but for which the sequence {fr} converges uniformly to a continuous function f on (0.1). 4. *Suppose thal f is uniformly continuous on R. For each n E N, sei fu(x) = f(x + ;). Prove that the sequence {fs} converges uniformly to f on R. 5. Let {fr} be a sequence of continuous real-valued functions that converges uniformly to a function f on a set EC R. Prove that lim fu(xn) = f(x) for every sequence {xn} C E such that Xn →x€ E. netic 8.2 Uniform Convergence 323 324 Chapter 8 Sequences and Series of Functions 8.2 Uniform Convergence f+€ All of the examples in the previous section show that pointwise convergence by itself is not sufficient to allow the interchange of limit operations; additional hypotheses are required. It was Weierstrass who, in the 1850s, realized what additional assumptions were needed to ensure that the limit function of a convergent sequence of continuous functions was again continuous. Recall from Definition 8.1.1, a sequence {fr} of real-valued functions defined on a set E converges pointwise to a function f on E if for each x E E. given e > 0, there ex- ists a positive integer ng n.(x, e) such that \$(.*) – f(x) 0, there exists a positive integer n, such that f(x) f(x) 0, there exists a positive integer no such that || fn – follow