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4130 HOMEWORK 8
Due Tuesday May 3
(1) Let
f
n
:
A
→
R
be functions which converge uniformly on
A
to a function
f
. Let
x
0
be
a cluster point of
A
. Suppose lim
x
→
x
0
f
n
(
x
) exists for all
n
. Let
L
n
= lim
x
→
x
0
f
n
(
x
).
(a) Show that the sequence
{
L
n
}
converges.
We show that
{
L
n
}
is a Cauchy sequence.
Let
ε >
0.
Since the sequence
{
f
n
}
converges uniformly, it is uniformly Cauchy, and so
there exists
N
∈
N
such that if
m,n > N
then

f
n
(
x
)

f
m
(
x
)

< ε/
3 for all
x
∈
A
.
Let
m,n > N
.
There exists
δ
1
>
0 such that if

x

x
0

< δ
1
, we have

f
n
(
x
)

L
n

< ε/
3.
There exists
δ
2
>
0 such that if

x

x
0

< δ
2
, we have

f
m
(
x
)

L
m

< ε/
3.
By the triangle inequality:

L
n

L
m
 ≤ 
L
n

f
n
(
x
)

+

f
n
(
x
)

f
m
(
x
)

+

f
m
(
x
)

L
m

for all
x
∈
A
. In particular, we can choose some
x
with

x

x
0

< δ
1
,δ
2
. Then
we get

L
n

L
m

< ε.
Therefore, for all
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