6. Let
{
f
n
}
∞
n
=1
be a sequence of real valued functions on a subset
E
of
R
.
Define the following sets: For each triple
(
n, m, k
)
∈
N
3
define
F
n,m,k
=
{
x
∈
E
:

f
n
(
x
)

f
m
(
x
)

<
1
k
}
.
For each pair (
N, k
)
∈
N
2
define
G
N,k
=
\
n
≥
N,m
≥
N
F
m,n,k
.
For each
k
∈
N
, define
H
k
=
∞
[
N
=1
G
N,k
. Finally, define
Z
=
∞
\
k
=1
H
k
.
Prove:
x
∈
Z
if and only if the sequence of real numbers
{
f
n
(
x
)
}
converges.
Solution.
Assume first
x
∈
Z
. We prove
{
f
n
(
x
)
}
is a Cauchy sequence of real numbers. For this let
² >
0 be given.
Let
k
∈
N
be such that 1
/k < ²
. Since
x
∈
Z
,
x
∈
H
k
by the definition of
Z
. By the definition of
H
k
, there is
N
∈
N
such that
x
∈
G
N,k
. By the definition of
G
N,k
,
x
∈
F
n,m,k
for all
n, m
≥
N
, thus

f
n
(
x
)

f
m
(
x
)

<
1
/k < ²
if
n, m
≥
N
.
It follows the sequence is Cauchy, hence converges. Conversely, let
x
∈
R
be such that
{
f
n
(
x
)
}
converges. Then it is
Cauchy and for every
k
∈
N
there is
N
∈
N
such that

f
n
(
x
)

f
m
(
x
)

<
1
/k
if
n, m
≥
N
, thus there is
N
∈
N
such
that
x
∈
F
n,m,k
for
n, m
≥
N
, hence
x
∈
G
N,k
. This proves that for every
k
∈
N
, there is
N
∈
N
such that
x
∈
G
N,k
,
thus for every
k
∈
N
,
x
∈
S
∞
N
=1
G
N,k
=
H
k
. Thus
x
∈
H
k
for all
k
, hence
x
∈
Z
.
2