Math 5215 - Supplementary Exercises Chapter 2 (Chapter 7 [Ru])
1. (a) Prove that x/n → 0 uniformly, as n → ∞, on any closed interval [a, b].
(b) Prove that 1/(nx) → 0 pointwise but not uniformly on (0, 1) as n → ∞.
2. (cf. Problem 1, Chapter 7, [Ru]) A sequence of functions fn is said to be uniformly bounded on a
set E if and only if there is an M > 0 such that |fn (x)| ≤ M for all x ∈ E and all n ∈ N.
Suppose that for each n ∈ N, fn : E → R is bounded. If fn → f uniformly on E, as n → ∞, prove
that {fn } is uniformly bounded on E and f is a bounded function on E.
3. Let [a, b] be a closed bounded interval, f : [a, b] → R be bounded, and g : [a, b] → R be continuous
with g(a) = g(b) = 0. Let fn be a uniformly bounded sequence of functions on [a, b].
Prove that if fn → f uniformly on all closed intervals [c, d] ⊂ (a, b), then fn g → f g uniformly on [a, b].
4. (cf. Problems 2, 3, Chapter 7, [Ru]) Suppose that fn → f and gn → g, as n → ∞, uniformly on
some set E ⊆ R.
(a) Prove that fn + gn → f + g and αfn → αf , as n → ∞, uniformly on E for all α ∈ R.
(b) Prove that fn gn → f g pointwise on E.
(c) Prove that if f and g are bounded on E, then fn gn → f g uniformly on E.
(d) Show that c) may be false when g is unbounded.
5. Suppose that E is a nonempty subset of R and that fn → f uniformly on E. Prove that if each fn
is uniformly continuous on E, then f is uniformly continuous on E.
6. Suppose that f is uniformly continuous on R. If yn → 0 as n → ∞ and fn (x) := f (x + yn ) for
x ∈ R, prove that fn converges uniformly on R.
7. Let f, g be continuous on a closed bounded interval [a, b] with |g(x)| > 0 for x ∈ [a, b]. Suppose that
fn → f and gn → g as n → ∞, uniformly on [a, b].
(a) Prove that 1/gn is defined for large n and fn /gn → f /g uniformly on [a, b] as n → ∞.
(b) Show that a) is false if [a, b] is replaced by (a, b).
8. Suppose that b > a > 0. Prove that
Z b
x n −x
e dx = b − a
lim
1+
n→∞ a
n
9. ProveZthat the following limits exist and evaluate them.
3
nx99 + 5
(a) lim
dx
n→∞ 1 x3 + nx66
Z 2
2
(b) lim
ex /n dx
n→∞ 0
Z 3r
x
(c) lim
sin + x + 1dx
n→∞ 0
n
1
10. Let fn be integrable on [0, 1] and fn → f uniformly on [0, 1]. Show that if bn ↑ 1 as n → ∞, then
Z
Z
1
f (x)dx
fn (x)dx =
lim
n→∞
bn
0
0
In the following problems, we always assume a, b ∈ R and a < b.
11. Let {fn } be a sequence in C[a, b]. Suppose that for all n ∈ N, fn is differentiable on (a, b) and
there exists M > 0 such that
|fn0 (x)| ≤ M, ∀x ∈ (a, b), ∀n ∈ N.
(1)
(a) Prove that the sequence {fn } is equicontinuous on [a, b].
(b) Assume, in addition to (1), that {fn (a)} is a bounded sequence (of real numbers). Prove that {fn }
is uniformly bounded on [a.b].
(c) Find an example of a sequence {fn } in C[0, 1] satisfying condition (1) such that {fn } is not pointwise
bounded at any point of [0, 1].
12. Let α > 0. A function f : [a, b] → R is said to be Hölder continuous of exponent α iff
|f (x) − f (x0 )|
0
0
Nα (f ) := sup
: x, x ∈ [a, b], x 6= x < ∞,
|x − x0 |α
|f (x) − f (x0 )|
0
0
: x, x ∈ [a, b], x 6= x is bounded above.
that is, the set
|x − x0 |α
(a) Prove that if f is Hölder continuous of exponent α then f is uniformly continuous on [a, b].
(b) Let {fn } be a sequence in C[a, b]. Suppose that there exists M > 0 such that
Nα (fn ) ≤ M, ∀n ∈ N.
Prove that the sequence {fn } is equicontinuous on [a, b].
13. Let {fn } be a sequence in C[0, 1] such that {fn } is pointwise bounded at all rational numbers in
[0, 1], that is, {fn (q)}n∈N is a bounded sequence (of numbers) for every q ∈ Q ∩ [0, 1]. Prove that there
exists a subsequence {fnk } of {fn } such that for all q ∈ Q ∩ [0, 1], {fnk (q)} is convergent.
14. In each of the following questions, contruct a sequence {fn } in C[0, 1] that satisfies all requirements
in the questions. Give full justifications for all your claims.
(a) {fn } is pointwise convergent on [0, 1], but not uniformly convergent on [0, 1].
(b) {fn } is pointwise convergent on [0, 1], but not uniformly bounded on [0, 1].
(c) {fn } is equicontinuous on [0, 1], but {fn } but is not pointwise convergent at any point of [0, 1].
(d) {fn } is uniformly bounded and equicontinuous on [0, 1], but {fn } is not uniformly convergent on
[0, 1].
(e) {fn } is uniformly bounded on [0, 1], but {fn } does not contain any subsequent that converges
uniformly on [0, 1].
15. Let A ⊂ C[a, b] (A denotes the uniformly closure of A).
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(a) Let f ∈ C[a, b]. Prove that f ∈ A if and only if for all > 0, there exists g ∈ A such that
sup |f (x) − g(x)| ≤ .
x∈[a,b]
(b) Prove that A is uniformly closed.
(c) Prove that A is uniformly closed if and only if A = A.
(d) Prove that if A is an algebra then A is also an algebra.
16. Let P be the set of all (real) polynomials, that is,
(
P =
p : R → R : ∃c0 , . . . , cn ∈ R : p(x) =
n
X
)
ck xk , ∀x ∈ R ,
k=0
Define
E[a, b] = {f : [a, b] → R : ∃p ∈ P : f (x) = p(ex ), ∀x ∈ [a, b]},
P
that is, f ∈ E[a, b] if and only if there are c0 , . . . , cn ∈ R such that f (x) = nk=0 ck ekx , ∀x ∈ [a, b].
(a) Prove that E[a, b] ⊂ C[a, b] and E[a, b] is an algebra.
(b) Prove that E[a, b] = C[a, b].
17. For P given in Problem 6, let
P0 = {p ∈ P : p(0) = 0} .
Define
P0 [a, b] = {f : [a, b] → R : ∃p ∈ P0 : f (x) = p(x), ∀x ∈ [a, b]}.
(a) Prove that f ∈ P0 [a, b] if and only if there are c1 , . . . cn ∈ R such that f (x) =
n
X
ck xk , ∀x ∈ [a, b].
k=1
(b) Prove that P0 [a, b] ⊂ C[a, b] and P0 [a, b] is an algebra.
(c) Assume 0 < a(< b). Using the Stone–Weierstrass theorem to prove that P0 [a, b] = C[a, b].
(d) Assume 0 ∈ [a, b]. Prove that P0 [a, b] 6= C[a, b]. What assumption of the Stone–Weierstrass theorem
is not satisfied in this case? Justify your answer.
18. Define
n
T [a, b] = f : [a, b] → R : ∃a0 , a1 , . . . an , b1 , . . . , bn ∈ R :
n
o
X
f (x) = a0 +
[ak cos(kx) + bk sin(kx)], ∀x ∈ [a, b] ,
k=1
(functions in T [a, b] are called trigonometric polynomials).
(a) Prove that T [a, b] ⊂ C[a, b] and T [a, b] is an algebra.
(b) Using the Stone–Weierstrass theorem to prove that T [0, π] = C[0, π].
(c) Prove that T [0, 2π] 6= C[0, 2π]. What assumption of the Stone–Weierstrass theorem is not satisfied
in this case? Justify your answer.
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