MATH185: Solutions to Assignment 10
1. Recall that a sequence of functions
f
n
:
U
→
C
is uniformly Cauchy
on
U
iﬀ
lim
N
→∞
(
sup
{
f
n
(
w
)

f
m
(
w
)

:
n,m
≥
N
and
w
∈
U
}
)
= 0
(a) Show that a sequence of functions
f
n
is Uniformly Cauchy on
U
if and only if there is a function
g
:
U
→
C
such that
f
n
converges
uniformly to
g
on
U
.
Sol:
Assume that (
f
n
)
n
is a uniformly Cauchy sequence of func
tions. For any
w
∈
U
, we have
(
*
)

f
n
(
w
)

f
m
(
w
)
 ≤
sup
{
f
n
(
z
)

f
m
(
z
)

:
z
∈
U
}
as a result, the sequence of complex numbers (
f
n
(
w
)) is a Cauchy
sequence and hence converges to a complex number which we will
call
g
(
w
). We will now show that the sequence of functions (
f
n
)
converges uniformly to the function
g
on
U
. Given
± >
0 there is
a number
N
such that, for all
n,m
≥
N
sup
{
f
n
(
z
)

f
m
(
z
)

:
z
∈
U
} ≤
±
so, by (
*
) we know that for all
n,m
≥
N
and all
w
∈
U

f
n
(
w
)

f
m
(
w
)
 ≤
±
taking limit when
m
→ ∞
we have that for all
w
∈
U

f
n
(
w
)

g
(
w
)
 ≤
±
and thus
sup
{
f
n
(
w
)

g
(
w
)

:
w
∈
U
} ≤
±
Since
±
was arbitrary it follows that
lim
n
→∞
sup
{
f
n
(
z
)

g
(
z
)

:
z
∈
U
}
= 0
so that (
f
n
)
n
converges uniformly to
g
on
U
.
Conversely, suppose that (
f
n
)
n
converges uniformly to
g
on
U
and let
± >
0 be given. Then there is an
N
∈
N
such that for
1
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≥
N sup
{
f
n
(
z
)

g
(
z
)

:
z
∈
U
}
<
±
2
. If
m,k
≥
N
we see that
for all
w
∈
U

f
n
(
w
)

f
m
(
w
)

=

f
n
(
w
)

g
(
w
)+
g
(
w
)

f
m
(
w
)
 ≤ 
f
n
(
w
)

g
(
w
)

+

g
(
w
)

f
m
(
w
)
 ≤
±
so
sup
{
f
n
(
w
)

f
m
(
w
)

:
n,m
≥
N
≤
±
. Since
±
was arbitrary it
follows that the sequence of functions (
f
n
)
n
is uniformly Cauchy
on
U
.
(b) Use part (
a
) to show that, if the series
∑
∞
n
=1

f
n
(
z
)

converges
uniformly on
U
then the series
∑
∞
n
=1
f
n
(
z
) converges uniformly
on
U
(Recall that a series converges unformly if it converges to
and moreover the sequence of partial sums converges uniformly
to the limit).
Sol:
Let
F
N
(
z
) =
∑
N
k
=1
f
n
(
z
) and let
A
N
(
z
) =
∑
N
k
=1

f
n
(
z
)

.
Saying that the series
∑
∞
n
=1

f
n
(
z
)

converges uniformly on
U
means that the sequence of partial sums (
A
N
)
N
converges uni
formly on
U
and thus, by part (a), that this sequence is uniformly
Cauchy on
U
.
We will now show that the sequence (
F
N
)
N
is uniformly Cauchy
on
U
. Assume
m
≥
n
, and let
w
∈
U

F
m
(
w
)

F
n
(
w
)

=

m
X
k
=
n
+1
f
k
(
w
)
 ≤
m
X
k
=
n
+1

f
k
(
w
)

=

A
m
(
w
)

A
n
(
w
)

In turn, for all
w
∈
U

A
m
(
w
)

A
n
(
w
)
 ≤
sup
{
A
m
(
z
)

A
n
(
z
)

:
z
∈
U
}
So putting both inequalities together we get
sup
{
F
m
(
z
)

F
n
(
z
)

:
z
∈
U
} ≤
sup
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 Fall '07
 Lim
 Math, Calculus, Uniform convergence, Modes of convergence, uniformly Cauchy sequence

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