201A, Fall ’10, Thomases
Homework 4  Solutions
Mihaela Ifrim
1. Suppose
f
n
∈
C
([0
,
1]) is a monotone decreasing sequence that con
verges pointwise to
f
∈
C
([0
,
1]). Prove that
f
n
converges uniformly
to
f
. This result is called
Dini’ s monotone convergence theorem
.
Proof.
•
Put
g
n
(
x
) :=
f
n
(
x
)

f
(
x
). The reason I am doing this is
because it will be easier for me to prove that
g
n
converges uniformly
to zero rather than prove that
f
n
converges uniformly to
f
(note that
this is just a rewriting of the our problem). The new sequence we have
constructed
g
n
is in
C
([0
,
1]) since
f
n
∈
C
([0
,
1]) and
f
∈
C
([0
,
1]).
Also, note that
g
n
→
0 pointwise and also
g
n
≥
g
n
+1
.
This last inequality is true since we know that
f
n
≥
f
n
+1
, which im
plies that
f
n

f
≥
f
n
+1

f
i.e.,
g
n
≥
g
n
+1
.
•
So, our goal is to prove that
g
n
→
0 uniformly on [0
,
1].
Let
ε >
0. For each
x
∈
[0
,
1] there is an integer
N
x
such that
0
≤
g
N
x
(
x
)
≤
ε
2
.
By the continuity and by the monotocity of the sequence
{
g
n
}
n
(we
said above that a monotone decreasing sequence), there exists an open
set
I
(
x
) that contains
x
, such that
0
≤
g
n
(
t
)
≤
ε
(1)
if
t
∈
I
(
x
) and if
n
≥
N
x
. Since [0
,
1] is compact, we know that we
can ﬁnd a ﬁnite open cover of it; in particular there exists a ﬁnite set
of points
x
1
, x
2
···
,x
m
such that
[0
,
1]
⊂
I
(
x
1
)
∪
I
(
x
2
)
∪
I
(
x
2
)
∪ ··· ∪
I
(
x
m
)
.
(2)
Choosing
N
:= max
{
N
x
1
,N
x
2
,N
x
3
,
···
,N
x
m
}