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3.42. 12. Suppose g and Saln= 1, 2, 3, ...) are defined on (0, o), are Riemann-integrable on [t, T] whenever 0

3.42. 12. Suppose g and Saln= 1, 2, 3, ...) are defined on (0, o), are Riemann-integrable on [t, T] whenever 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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Q12 real analysis by Walton rudin
function f on a set E. Prove that lim f.(x,) = f(x) for every sequence of points x, e E such that x, x, and x e E, Is the converse of this true? SEQUENCES AND SERIES OF FUNCTIONS 167 10. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4, for the definition), consider the function (nx) f(x) = (x real). Find all discontinuities of f, and show that they form a countable dense set. Show that fis nevertheless Riemann-integrable on every bounded interval. 11. Suppose {fa}, {gn) are defined on E, and (a) Ef, has uniformly bounded partial sums; (b) g, →0 uniformly on E; (c) g.(x) 29:(x)Z 9s(x) 2·… for every x € E. Prove that E fng, converges uniformly on E. Hint: Compare with Theorem 3.42. 12. Suppose g and f.(n= 1, 2, 3, ...) are defined on (0, ∞), are Riemann-integrable on [t, T] whenever 0<t<T<∞, |fal <9,f.→f uniformly on every compact sub- set of (0, o), and g(x) dx < 0. Prove that m fx) dx= lim (See Exercises 7 and 8 of Chap. 6 for the relevant definitions.) This is a rather weak form of Lebesgue's dominated convergence theorem (Theorem 11.32). Even in the context of the Riemann integral, uniform conver- gence can be replaced by pointwise convergence if it is assumed that fE R. (See the articles by F. Cunningham in Math. Mag., vol. 40, 1967, pp. 179–186, and by H. Kestelman in Amer. Math. Monthly, vol. 77, 1970, pp. 182–187.) 13. Assume that {f) is a sequence of monotonically increasing functions on R' with OSA(x)S1 for all x and all n. (a) Prove that there is a function f and a sequence {n} such that f(x) = lim fa,(x) for every x e R'. (The existence of such a pointwise convergent subsequence is usually called Helly's selection theorem.) (b) If, moreover, f is continuous, prove that fa f uniformly on compact sets. Hin!: (i) Some subsequence {fa} converges at all rational points r, say, to f(t). (ii) Define fS(x), for any x e R', to be sup f(r), the sup being taken over all rSx. (ii) Show that fa(x) →f(x) at every x at which f is continuous. (This is where monotonicity is strongly used.) (iv) A subsequence of {fa} converges at every point of discontinuity of f since there are at most countably many such points. This proves (a). To prove (b), modify your proof of (iii) appropriately. 168 PRINCIPLES OF MATHEMATICAL ANALYSIS 14. Let f be a continuous real function on R' with the following properties: Os S(1)<1, f(t+ 2) = f(t) for every t, and (0
Transcribed Image Text:function f on a set E. Prove that lim f.(x,) = f(x) for every sequence of points x, e E such that x, x, and x e E, Is the converse of this true? SEQUENCES AND SERIES OF FUNCTIONS 167 10. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4, for the definition), consider the function (nx) f(x) = (x real). Find all discontinuities of f, and show that they form a countable dense set. Show that fis nevertheless Riemann-integrable on every bounded interval. 11. Suppose {fa}, {gn) are defined on E, and (a) Ef, has uniformly bounded partial sums; (b) g, →0 uniformly on E; (c) g.(x) 29:(x)Z 9s(x) 2·… for every x € E. Prove that E fng, converges uniformly on E. Hint: Compare with Theorem 3.42. 12. Suppose g and f.(n= 1, 2, 3, ...) are defined on (0, ∞), are Riemann-integrable on [t, T] whenever 0<t<T<∞, |fal <9,f.→f uniformly on every compact sub- set of (0, o), and g(x) dx < 0. Prove that m fx) dx= lim (See Exercises 7 and 8 of Chap. 6 for the relevant definitions.) This is a rather weak form of Lebesgue's dominated convergence theorem (Theorem 11.32). Even in the context of the Riemann integral, uniform conver- gence can be replaced by pointwise convergence if it is assumed that fE R. (See the articles by F. Cunningham in Math. Mag., vol. 40, 1967, pp. 179–186, and by H. Kestelman in Amer. Math. Monthly, vol. 77, 1970, pp. 182–187.) 13. Assume that {f) is a sequence of monotonically increasing functions on R' with OSA(x)S1 for all x and all n. (a) Prove that there is a function f and a sequence {n} such that f(x) = lim fa,(x) for every x e R'. (The existence of such a pointwise convergent subsequence is usually called Helly's selection theorem.) (b) If, moreover, f is continuous, prove that fa f uniformly on compact sets. Hin!: (i) Some subsequence {fa} converges at all rational points r, say, to f(t). (ii) Define fS(x), for any x e R', to be sup f(r), the sup being taken over all rSx. (ii) Show that fa(x) →f(x) at every x at which f is continuous. (This is where monotonicity is strongly used.) (iv) A subsequence of {fa} converges at every point of discontinuity of f since there are at most countably many such points. This proves (a). To prove (b), modify your proof of (iii) appropriately. 168 PRINCIPLES OF MATHEMATICAL ANALYSIS 14. Let f be a continuous real function on R' with the following properties: Os S(1)<1, f(t+ 2) = f(t) for every t, and (0
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