1. Exercise 6.2.16. For each
n
∈
N
, let
f
n
be a function deﬁned on [0
,
1]. If (
f
n
) is bounded on
[0
,
1]—that is, there exists an
M >
0 such that

f
n
(
x
)
 ≤
M
for all
n
∈
N
and
x
∈
[0
,
1]—and
if the collection of functions (
f
n
) is equicontinuous (Exercise 6.2.15), follow these steps to
show that (
f
n
) contains a uniformly convergent subsequence.
(a) Use Exercise 6.2.14 to produce a subsequence (
f
n
k
) that converges at every rational point
in [0
,
1]. To simplify the notation, set
g
k
=
f
n
k
. It remains to show that (
g
k
) converges
uniformly on all of [0
,
1].
Proof.
Since (
f
n
) is bounded and
Q
∩
[0
,
1] is countable, Exercise 6.2.14 implies that
there is a subsequence (
f
n
k
) that converges at all points in
Q
∩
[0
,
1].
(b) Let
± >
0. By equicontinuity, there exists a
δ >
0 such that
g
k
(
x
)

g
k
(
y
)

<
±
3
for all

x

y

< δ
and
k
∈
N
. Using this
δ
, let
r
1
, r
2
, . . . , r
m
be a
ﬁnite
collection of
rational points with the property that the union of the neighborhoods
V
δ
(
r
i
) contains
[0
,
1].
Explain why there must exist an
N
∈
N
such that

g
s
(
r
i
)

g
t
(
r
i
)

<
±
3
for all
s, t
≥
N
and
r
i
in the ﬁnite subset of [0
,
1] just described. Why does having the
set
{
r
1
, r
2
, . . . , r
m
}
be ﬁnite matter?
Proof.
Let
± >
0. Since
r
1
, . . . , r
m
∈
Q
∩
[0
,
1], we know that each sequence (
g
k
(
r
i
))
converges and, hence, is Cauchy. Therefore, for each
i
= 1
, . . . , m
, there exists
N
i
∈
N
such that
r, s
≥
N
i
implies that

g
s
(
r
i
)

g
t
(
r
i
)

<
±
3
.
Let
N
= max
{
N
1
, . . . , N
m
}
(this is where it’s essential that the collection
{
r
1
, . . . , r
m
}
is ﬁnite; if it were inﬁnite, then there might well not be a maximum
N
i
). Then, for any
r, s
≥
N
and for any
i
∈ {
1
, . . . , m
}
, we have that
s, t
≥
N
i
and so

g
s
(
r
i
)

g
t
(
r
i
)

<
±
3
,
as desired.
(c) Finish the argument by show that, for an arbitrary
x
∈
[0
,
1],

g
s
(
x
)

g
t
(
x
)

< ±
for all
s, t
≥
N
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Math, Calculus, Mean Value Theorem, Cauchy, Uniform convergence, Hn, lim Gn

Click to edit the document details