Solutions to homework # 13.
1. First of all, for every function
f
n
there exists a bound
M
n
such that

f
n
(
x
)
 ≤
M
n
for all
x
. Now pick an arbitrary
ε >
0. Then there exists
N
=
N
(
ε
) such that

f
n
(
x
)

f
N
(
x
)

< ε
for all
x
whenever
n > N.
Then

f
n
(
x
)
 ≤
ε
+

f
N
(
x
)
 ≤
M
N
+
ε
for all
n
≥
N
. Let
M
:= max
{
M
1
, . . . , M
N

1
, M
N
+
ε
}
.
Then

f
n
(
x
)
 ≤
M
for all
n
and all
x
.
2.
Let
f
denote the limit of (
f
n
) and let
g
denote the limit if (
g
n
). For any
ε >
0, there
exists
N
=
N
(
ε
) such that

f
n
(
x
)

f
(
x
)
 ≤
ε/
2,

g
n
(
x
)

g
(
x
)
 ≤
ε/
2 for all
x
. Then

f
n
(
x
) +
g
n
(
x
)

f
(
x
)

g
(
x
)
 ≤ 
f
n
(
x
)

f
(
x
)

+

g
n
(
x
)

g
(
x
)
 ≤
ε
forall
x.
If now both (
f
n
) and (
g
n
) are sequences of bounded functions, we know from Exercise 1
that they are uniformly bounded. So there exists a constant such that

f
n
(
x
)
 ≤
M
,

g
n
(
x
)
 ≤
M
for all
x
and all
n
∈
IN. Then the limits
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This document was uploaded on 11/17/2010.
 Spring '09

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