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
n

x

< ε,
by defn of convergence. Thus, all points save
{
x
1
, . . . , x
N

1
}
must lie in the interval.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 Wong
 Limits, Limit of a function, Supremum, Limit of a sequence, Limit superior and limit inferior, Xn, subsequence

Click to edit the document details