FUNDAMENTALS OF ANALYSIS
W W L CHEN
c
W W L Chen, 1983, 2008.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for financial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
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Chapter 8
UNIFORM CONVERGENCE
8.1. Introduction
We begin by making a somewhat familiar definition.
Definition. Suppose that fn : X → C is a sequence of functions on a set X ⊆ R. We say that the
sequence fn converges pointwise to the function f : X → C if for every x ∈ X, we have
|fn (x) − f (x)| → 0
as n → ∞.
Example 8.1.1. Let X = [0, 1]. For every n ∈ N and every x ∈ [0, 1], let fn (x) = xn . Then for every
x ∈ [0, 1], fn (x) → f (x) as n → ∞, where f (x) = 0 if 0 ≤ x < 1 and f (1) = 1. Note that each of the
functions fn (x) is continuous on [0, 1], but the limit function f (x) is not continuous on [0, 1]. Hence the
continuity property of the functions fn (x) is not carried over to the limit function f (x).
To carry over certain properties of the individual functions of a sequence to the limit function, we
need a type of convergence which is stronger than pointwise convergence.
Definition. Suppose that fn : X → C is a sequence of functions on a set X ⊆ R. We say that the
sequence fn converges uniformly to the function f : X → C if
sup |fn (x) − f (x)| → 0
as n → ∞.
x∈X
Example 8.1.2. In Example 8.1.1, we have fn (x) → f (x) pointwise in [0, 1]. However, if 0 ≤ x < 1,
then |fn (x) − f (x)| = xn and so
sup |fn (x) − f (x)| ≥ sup |fn (x) − f (x)| = sup xn = 1
x∈[0,1]
x∈[0,1)
x∈[0,1)
for every n ∈ N. It follows that fn (x) → f (x) as n → ∞, pointwise but not uniformly on [0, 1].
Chapter 8 : Uniform Convergence
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Remark. Pointwise convergence means that given any > 0, for every x ∈ X, there exists N = N (, x)
such that
|fn (x) − f (x)| < whenever n > N (, x).
Uniform convergence means that given any > 0, there exists N = N (), independent of x ∈ X, such
that
|fn (x) − f (x)| < whenever n > N () and x ∈ X.
8.2. Criteria for Uniform Convergence
We shall first of all extend the General principle of convergence to the case of uniform convergence.
THEOREM 8A. (GENERAL PRINCIPLE OF UNIFORM CONVERGENCE) Suppose that fn is a
sequence of real or complex valued functions defined on a set X ⊆ R. Then fn (x) converges uniformly
on X as n → ∞ if and only if, given any > 0, there exists N such that
sup |fm (x) − fn (x)| < whenever m > n ≥ N .
x∈X
Proof. (⇒) Suppose that fn (x) → f (x) uniformly on X as n → ∞. Then given any > 0, there exists
N such that
sup |fn (x) − f (x)| < 12 whenever n ≥ N .
x∈X
It follows that
|fm (x) − fn (x)| ≤ |fm (x) − f (x)| + |fn (x) − f (x)| < whenever m > n ≥ N and x ∈ X,
and so
sup |fm (x) − fn (x)| ≤ whenever m > n ≥ N .
x∈X
(⇐) Since R and C are complete, for every x ∈ X, the sequence fn (x) converges pointwise to a limit
f (x), say, as n → ∞. We shall show that fn (x) → f (x) uniformly on X as n → ∞. Given any > 0,
there exists N such that for every x ∈ X,
|fm (x) − fn (x)| < whenever m > n ≥ N .
Hence for every x ∈ X,
|f (x) − fn (x)| = lim |fm (x) − fn (x)| ≤ m→∞
whenever n ≥ N ,
so that
sup |fn (x) − f (x)| ≤ whenever n ≥ N .
x∈X
Hence fn (x) → f (x) uniformly on X as n → ∞. We next turn our attention to series of real or complex valued functions.
Chapter 8 : Uniform Convergence
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Definition. Suppose that un is a sequence of real or complex valued functions defined on a set X ⊆ R.
We say that the series
∞
X
un (x)
n=1
converges uniformly on X if the sequence of partial sums
N
X
sN (x) =
un (x)
n=1
converges uniformly on X.
We have the analogue of the Comparison test.
THEOREM 8B. (WEIERSTRASS’S M-TEST) Suppose that un is a sequence of real or complex valued
functions defined on a set X ⊆ R. Suppose further that for every n ∈ N, there exists a real constant Mn
such that the series
∞
X
Mn
n=1
is convergent, and that |un (x)| ≤ Mn for every x ∈ X. Then the series
∞
X
un (x)
n=1
converges uniformly and absolutely on X.
Proof. Given any > 0, it follows from the General principle of convergence for series that there exists
N such that
whenever m > n ≥ N .
Mn+1 + . . . + Mn < It follows that
|sm (x) − sn (x)| ≤ Mn+1 + . . . + Mn < whenever m > n ≥ N and x ∈ X,
so that
sup |sm (x) − sn (x)| ≤ whenever m > n ≥ N .
x∈X
It now follows from Theorem 8A that the series
∞
X
un (x)
n=1
converges uniformly on X. Note finally that absolute convergence follows pointwise from the proof of
the Comparison test. The General principle of uniform convergence can also be used to establish the following two results.
Chapter 8 : Uniform Convergence
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THEOREM 8C. (DIRICHLET’S TEST) Suppose that an and bn are two sequences of real valued
functions defined on a set X ⊆ R, and satisfy the following conditions:
(a) There exists K ∈ R such that |sn (x)| ≤ K for every n ∈ N and every x ∈ X, where sn (x) denotes
the sequence of partial sums sn (x) = a1 (x) + . . . + an (x).
(b) For every x ∈ X, the sequence bn (x) is monotonic.
(c) The sequence bn (x) → 0 uniformly on X as n → ∞.
Then the series
∞
X
an (x)bn (x)
n=1
converges uniformly on X.
Proof. Since bn (x) → 0 uniformly on X as n → ∞, given any > 0, there exists N0 such that
|bn (x)| <
4K
whenever n > N0 and x ∈ X.
It follows that whenever M > N ≥ N0 , we have
M
X
an (x)bn (x) = |(sN +1 (x) − sN (x))bN +1 (x) + . . . + (sM (x) − sM −1 (x))bM (x)|
n=N +1
= | − sN (x)bN +1 (x) + sN +1 (x)(bN +1 (x) − bN +2 (x)) + . . . + sM −1 (x)(bM −1 (x) − bM (x)) + sM (x)bM (x)|
≤ K(|bN +1 (x)| + |bN +1 (x) − bN +2 (x)| + . . . + |bM −1 (x) − bM (x)| + |bM (x)|)
= K(|bN +1 (x)| + |bN +1 (x) − bM (x)| + |bM (x)|) ≤ 2K(|bN +1 (x)| + |bM (x)|) < .
The result follows from the General principle of uniform convergence. THEOREM 8D. (ABEL’S TEST) Suppose that an and bn are two sequences of real valued functions
defined on a set X ⊆ R, and satisfy the following conditions:
∞
X
(a) The series
an (x) converges uniformly on X.
n=1
(b) For every x ∈ X, the sequence bn (x) is monotonic.
(c) There exists K ∈ R such that |bn (x)| ≤ K for every n ∈ N and every x ∈ X.
Then the series
∞
X
an (x)bn (x)
n=1
converges uniformly on X.
Proof. Given any > 0, there exists N0 such that
m
X
an (x) <
3K
whenever m > N ≥ N0 and x ∈ X.
n=N +1
In other words, writing sn (x) = a1 (x) + . . . + an (x), we have
|sm (x) − sN (x)| <
Chapter 8 : Uniform Convergence
3K
whenever m > N ≥ N0 and x ∈ X.
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It follows that whenever M > N ≥ N0 , we have
M
M
X
X
(sm (x) − sm−1 (x))bm (x)
am (x)bm (x) = m=N +1
m=N +1
M
X
=
((sm (x) − sN (x)) − (sm−1 (x) − sN (x)))bm (x)
m=N +1
M
M
−1
X
X
=
(sm (x) − sN (x))bm (x) −
(sm (x) − sN (x))bm+1 (x)
m=N +1
M
−1
X
≤
m=N +1
|sm (x) − sN (x)||bm (x) − bm+1 (x)| + |sM (x) − sN (x)||bM (x)|
m=N +1
<
3K
=
3K
M
−1
X
|bm (x) − bm+1 (x)| +
m=N +1
|bM (x)|
3K
M −1
X
(bm (x) − bm+1 (x)) +
|bM (x)|
3K
m=N +1
=
|bN +1 (x) − bM (x)| +
|bM (x)|
3K
3K
≤
(|bN +1 (x)| + 2|bM (x)|) ≤ .
3K
The result follows from the General principle of uniform convergence. 8.3. Consequences of Uniform Convergence
In this section, we discuss the implications of uniform convergence on continuity, integrability and
differentiability. To answer the question first raised in Section 8.1, we have the following result.
THEOREM 8E. Suppose that a sequence of functions fn : X → C converges uniformly on a set X ⊆ R
to a function f : X → C as n → ∞. Suppose further that c ∈ X and that the function fn is continuous
at c for every n ∈ N. Then the function f is continuous at c.
Remark. The conclusion of Theorem 8E can be written in the form
lim lim fn (x) = lim lim fn (x).
x→c n→∞
n→∞ x→c
Theorem 8E then says that if the sequence of functions converges uniformly on X, then the order of the
two limiting processes can be interchanged.
Proof of Theorem 8E. Given any > 0, there exists n ∈ N such that
sup |fn (x) − f (x)| <
x∈X
.
3
Since fn is continuous at c, there exists δ > 0 such that
|fn (x) − fn (c)| <
3
whenever |x − c| < δ.
It follows that whenever |x − c| < δ, we have
|f (x) − f (c)| ≤ |f (x) − fn (x)| + |fn (x) − fn (c)| + |fn (c) − f (c)| < .
Hence f is continuous at c. Chapter 8 : Uniform Convergence
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We immediately have the following corollary of Theorem 8E.
THEOREM 8F. Suppose that un is a sequence of real or complex valued functions defined on a set
X ⊆ R, and that the series
∞
X
un (x)
n=1
converges uniformly to a function s(x) on X. Suppose further that c ∈ X and that the function un is
continuous at c for every n ∈ N. Then the function s is continuous at c.
We next study the effect of uniform convergence on integrability.
THEOREM 8G. Suppose that fn is a sequence of real valued functions integrable over a closed interval
[A, B]. Suppose further that fn → f uniformly on [A, B] as n → ∞. Then the function f is integrable
over [A, B], and
Z
B
Z
B
fn (x) dx.
f (x) dx = lim
n→∞
A
(1)
A
Remark. The conclusion of Theorem 8G can be written in the form
Z B
Z B
lim fn (x) dx = lim
fn (x) dx.
A
n→∞
n→∞
A
Theorem 8G then says that if the sequence of functions converges uniformly on [A, B], then the order
of integration and taking limits as n → ∞ can be interchanged.
Proof of Theorem 8G. Given any > 0, there exists N ∈ N such that
sup |fn (x) − f (x)| <
x∈[A,B]
3(B − A)
whenever n ≥ N .
(2)
It follows in particular that
fN (x) −
< f (x) < fN (x) +
3(B − A)
3(B − A)
whenever x ∈ [A, B].
Hence for any dissection ∆ of [A, B], we have
s(fN , ∆) −
≤ s(f, ∆) ≤ S(f, ∆) ≤ S(fN , ∆) + ,
3
3
so that
S(f, ∆) − s(f, ∆) ≤ S(fN , ∆) − s(fN , ∆) +
2
.
3
Since fN is integrable over [A, B], there exists a dissection ∆ of [A, B] such that
S(fN , ∆) − s(fN , ∆) <
,
3
so that
S(f, ∆) − s(f, ∆) < .
Hence f is integrable over [A, B]. On the other hand, it follows from (2) that
Z
Z
Z B
B
B
fn (x) dx −
f (x) dx ≤
|fn (x) − f (x)| dx < whenever n ≥ N .
A
A
A
The assertion (1) follows immediately. Chapter 8 : Uniform Convergence
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W W L Chen, 1983, 2008
We immediately have the following corollary of Theorem 8G.
THEOREM 8H. Suppose that un is a sequence of real valued functions defined on a closed interval
[A, B], and that the series
∞
X
un (x)
n=1
converges uniformly to a function s(x) on [A, B]. Suppose further that the function un is integrable over
[A, B] for every n ∈ N. Then the function s is integrable over [A, B], and
Z
B
s(x) dx =
A
∞ Z
X
B
un (x) dx.
A
n=1
Remark. The conclusion of Theorem 8H can be written in the form
!
Z B X
∞
∞ Z B
X
un (x) dx =
un (x) dx.
A
n=1
n=1
A
Theorem 8H then says that if the sequence of functions converges uniformly on [A, B], then the order of
integration and summation can be interchanged. In other words, the series can be integrated term by
term.
We next study the effect of uniform convergence on differentiability.
THEOREM 8J. Suppose that fn is a sequence of real valued functions differentiable in a closed interval
[A, B]; in other words, differentiable at every point in the open interval (A, B), right differentiable at A
and left differentiable at B. Suppose further that the sequence fn (x0 ) converges for some x0 ∈ [A, B],
and that the sequence fn0 converges uniformly on [A, B]. Then the sequence fn converges uniformly on
[A, B], and the limit function f is differentiable in [A, B]. Furthermore, for every x ∈ [A, B], we have
f 0 (x) = lim fn0 (x).
n→∞
Remark. The conclusion of Theorem 8J can be written in the form
0
lim fn (x) = lim fn0 (x).
n→∞
n→∞
Theorem 8J then says essentially that if the sequence of functions satisfies some mild convergence property and the sequence of derivatives converges uniformly on [A, B], then the order of differentiation and
taking limits as n → ∞ can be interchanged.
Proof of Theorem 8J. Suppose that fn0 → g as n → ∞. Since the convergence is uniform in [A, B],
given any > 0, there exists N such that
sup |fn0 (x) − g(x)| <
[A,B]
4(1 + (B − A))
whenever n ≥ N ,
(3)
so that
0
sup |fm
(x) − fn0 (x)| <
[A,B]
Chapter 8 : Uniform Convergence
2(1 + (B − A))
whenever m > n ≥ N .
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Suppose that η1 , η2 ∈ [A, B]. Applying the Mean value theorem to the function fm − fn , we have
0
|(fm (η1 ) − fn (η1 )) − (fm (η2 ) − fn (η2 ))| = |η1 − η2 ||fm
(ξ) − fn0 (ξ)|
< |η1 − η2 |
<
2(1 + (B − A))
2
(4)
for some ξ between η1 and η2 . On the other hand, since fn (x0 ) converges as n → ∞, there exists N 0
such that
|fm (x0 ) − fn (x0 )| <
4
whenever m > n ≥ N 0 .
It follows from (4), with η1 = x and η2 = x0 , that
|fm (x) − fn (x)| < |fm (x0 ) − fn (x0 )| +
3
<
2
4
whenever m > n ≥ max{N, N 0 },
and so it follows from the Principle of uniform convergence that fn (x) converges uniformly in [A, B].
Suppose that fn (x) → f (x) as n → ∞. Let c ∈ [A, B] be fixed. For every x ∈ [A, B], it follows from (4),
with η1 = x and η2 = c, that
fm (x) − fm (c) fn (x) − fn (c) < −
2
x−c
x−c
whenever m > n ≥ N ,
so that on letting m → ∞, we have
f (x) − f (c) fN (x) − fN (c) < .
−
x−c
2
x−c
(5)
Since fN is differentiable at c, there exists δ > 0 such that
fN (x) − fN (c)
0
− fN (c) <
x−c
4
whenever 0 < |x − c| < δ and x ∈ [A, B].
(6)
Combining (5), (6) and (3), we conclude that
f (x) − f (c)
− g(c)
x−c
f (x) − f (c) fN (x) − fN (c) fN (x) − fN (c)
0
0
≤
−
− fN (c) + |fN
+
(c) − g(c)| < x−c
x−c
x−c
whenever 0 < |x − c| < δ and x ∈ [A, B]. Hence
f 0 (c) = g(c) = lim fn0 (c).
n→∞
This completes the proof. We immediately have the following corollary of Theorem 8J.
Chapter 8 : Uniform Convergence
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W W L Chen, 1983, 2008
THEOREM 8K. Suppose that un is a sequence of real valued functions differentiable in a closed interval
[A, B]. Suppose further that the series
∞
X
un (x0 )
n=1
converges for some x0 ∈ [A, B], and that the series
∞
X
u0n (x)
n=1
converges uniformly on [A, B]. Then the series
∞
X
un (x)
n=1
converges uniformly on [A, B], and its sum s(x) is differentiable in [A, B]. Furthermore, for every
x ∈ [A, B], we have
s0 (x) =
∞
X
u0n (x).
n=1
Remark. The conclusion of Theorem 8K can be written in the form
!0
∞
∞
X
X
u0n (x).
un (x) =
n=1
n=1
Theorem 8K then says essentially that if the series of functions satisfies some mild convergence property and the series of derivatives converges uniformly on [A, B], then the order of differentiation and
summation can be interchanged.
8.4. Application to Power Series
Consider a power series in z ∈ C, of the form
∞
X
an z n ,
(7)
n=0
where an ∈ C for every n ∈ N ∪ {0}. Recall Theorem 3Q, that if the power series (7) has radius of
convergence R and if 0 < r < R, then the series
∞
X
|an |rn
n=0
converges. It follows from Weierstrass’s M-test that the power series (7) converges uniformly on the set
{z ∈ C : |z| ≤ r}. Suppose now that |z0 | < R. Then there exists r such that |z0 | < r < R. It follows
from Theorem 8F that the power series is continuous at z0 . We have therefore proved the following
result.
THEOREM 8L. Suppose that the power series (7) has radius of convergence R. Then for every r
satisfying 0 < r < R, the power series converges uniformly on the set {z ∈ C : |z| ≤ r}. Furthermore,
the sum of the power series is continuous on the set {z ∈ C : |z| < R}.
Chapter 8 : Uniform Convergence
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We next consider real power series.
THEOREM 8M. Suppose that the real power series
∞
X
an xn ,
(8)
n=0
where an ∈ R for every n ∈ N ∪ {0}, converges in the interval (−R, R) to a function f (x). Then f (x) is
differentiable on (−R, R), and
∞
X
f 0 (x) =
nan xn−1 .
n=1
On the other hand, if |X| < R, then
X
Z
f (x) dx =
0
∞
X
an
X n+1 .
n
+
1
n=0
Proof. It is not difficult to see that the power series
∞
X
nan xn−1
(9)
n=1
converges in the interval (−R, R). It follows from Theorem 8L that the series (9) converges uniformly on
any closed subinterval of (−R, R). The first assertion follows from Theorem 8K. The second assertion
follows from Theorem 8H on noting that the power series converges uniformly on the closed interval with
endpoints 0 and X. We conclude this chapter by establishing the following useful result.
THEOREM 8N. (ABEL’S THEOREM) Suppose that the real series
∞
X
an
n=0
is convergent. Then
∞
X
n=0
an xn →
∞
X
an
as x → 1 − .
n=0
Proof. It follows from Abel’s test that the series
∞
X
an xn
n=0
converges uniformly on [0, 1]. Let s(x) be its sum. Then it follows from Theorem 8F that s(x) is
continuous on [0, 1]. In particular, we have s(x) → s(1) as x → 1−. Chapter 8 : Uniform Convergence
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Problems for Chapter 8
1. For each of the following, prove that the sequence of functions converges pointwise on its domain of
definition as n → ∞, and determine whether the convergence is uniform on this set:
nx
nx
a) fn (x) =
on [0, ∞)
on [0, ∞)
b) fn (x) =
n+x
1 + n2 x2
sin nx
c) fn (x) = xn (1 − x) on [0, 1]
d) fn (x) =
on (0, 1)
nx
2
e) fn (x) = nxe−nx on [0, 1]
2. Suppose that fn and gn are complex valued functions defined on a set X ⊆ R. Suppose further that
fn (x) → f (x) and gn (x) → g(x) as n → ∞ uniformly on X.
a) Prove that αfn (x) + βgn (x) → αf (x) + βg(x) as n → ∞ uniformly on X for any α, β ∈ C.
b) Is it necessarily true that fn (x)gn (x) → f (x)g(x) as n → ∞ uniformly on X? Justify your
assertion.
3. a) Suppose that fn (x) → f (x) as n → ∞ uniformly on each of the sets X1 , . . . , Xk in R. Prove that
fn (x) → f (x) as n → ∞ uniformly on the union X1 ∪ . . . ∪ Xk .
b) Give an example to show that the analogue for an infinite collection of sets does not hold.
4. The series
∞
X
un (x) is uniformly convergent on a set S ⊆ R.
n=1
a) Is the series necessarily absolutely convergent for every x ∈ S? Justify your assertion.
b) Is the series necessarily absolutely convergent for some x ∈ S? Justify your assertion.
5. Prove that the series
6. Suppose that
∞
X
∞
X
(−1)n
converges uniformly on R.
n(1 + x2n )
n=1
an is a convergent real series.
n=1
a) Prove that the series
b) Prove that the series
∞
X
n=1
∞
X
an xn converges uniformly on [0, 1].
an n−x converges uniformly on [0, ∞).
n=1
7. For every n ∈ N, let fn (x) = n−1 e−x/n .
a) Show that fn (x) converges uniformly on (0, ∞).
Z ∞
Z ∞
b) Show that lim
fn (x) dx and
lim fn (x) dx both exist but are not equal.
n→∞
0
0
n→∞
xn
.
1 + x2n
a) For what values of x ∈ R does fn (x) converge? Find the limit function f (x) for these values.
b) Prove that fn (x) converges uniformly on any interval [A, B] in R such that
(i) [A, B] ⊆ (−∞, −1); or
(ii) [A, B] ⊆ (−1, 1); or
(iii) [A, B] ⊆ (1, ∞).
c) Can fn (x) converge uniformly on any interval I ⊆ R such that 1 ∈ I? Justify your assertion.
8. For every n ∈ N, let fn (x) =
Chapter 8 : Uniform Convergence
page 11 of 11