3.1 Limits and bounds
43
Example 3.1.4.
The sequence
{
(

1)
n
(1+
1
n
)
}
has the monotone decreasing subsequence
1 +
1
2
n
obtained by taking every second term. Inﬁnitely many deletions.
x
n
=

2
,
3
2
,

4
3
,
5
4
,

6
5
,
7
6
,
...
n
=
1
,
2
,
3
,
4
,
5
,
6
,
...
n
1
= 2
,
n
2
= 4
,
n
3
= 6
,
...
x
n
1
=
x
2
=
3
2
x
n
2
=
x
4
=
5
4
,
x
n
3
=
x
6
=
7
6
,
Note:
n
k
≥
k
.
Example 3.1.5.
The sequence
{
1
,
1
2
,
1
3
,
1
4
,...
}
has subsequence
{
1
2
,
1
3
,
1
4
,...
}
obtained by
deleting the ﬁrst term. A single deletion.
Theorem 3.1.13.
Let
{
x
n
}
be a sequence in
R
.
(i)
x
n
→
x
iﬀ every neighbourhood of
x
of the form
(
x

ε,x
+
ε
)
,ε >
0
contains all but
ﬁnitely many points
x
n
.
(ii)
x
∈
R
is a limit point of the set
A
iﬀ there is a sequence
{
x
n
} ⊆
A
with
x
n
→
x
and
x
n
6
=
x
.
(iii)
x
n
→
x
iﬀ
x
n
k
→
x
, for every subsequence
{
x
n
k
}
.
(i).
(
⇒
) Suppose
x
n
→
x
and ﬁx any
ε >
0. Corresponding to this
ε
, there is an
N
∈
N
such that
n
≥
N
=
⇒ 
x