11. Suppose (f.), {aa) are defined on E, and (a) Ef, has uniformly bounded partial sums; (b) g. --0 uniformly on E; (c) g.(x) 20:(x) (x) 2… for every x € E. Prove that E fagn converges uniformly on E.
11. Suppose (f.), {aa) are defined on E, and (a) Ef, has uniformly bounded partial sums; (b) g. --0 uniformly on E; (c) g.(x) 20:(x) (x) 2… for every x € E. Prove that E fagn converges uniformly on E.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Q11 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa29dd1fd-ff33-4092-aedd-03e5d354d4ee%2F602bc5d4-261c-4e60-a16d-6d08d49e4d68%2Fhopm7re_processed.jpeg&w=3840&q=75)
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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