THEOREM 6.
Suppose that
{
s
n
}
and
{
t
n
}
are sequences such that
s
n
≤
t
n
for all
n
.
1. If
s
n
→
+
∞
,
then
t
n
→
+
∞
.
2. If
t
n
→ ∞
,
then
s
n
→ ∞
.
THEOREM 7.
Let
{
s
n
}
be a sequence of positive numbers.
Then
s
n
→
+
∞
if and only if
1
/s
n
→
0
.
Proof:
Suppose
s
n
→ ∞
. Let
>
0
and set
M
= 1
/
. Then there exists a positive integer
N
such that
s
n
> M
for all
n > N
. Since
s
n
>
0,
1
/s
n
<
1
/M
=
for all
n > N
which implies
1
/s
n
→
0.
Now suppose that
1
/s
n
→
0. Choose any positive number
M
and let
= 1
/M
. Then there
exists a positive integer
N
such that
0
<
1
s
n
<
=
1
M
for all
n > N.
that is,
1
s
n
<
1
M
.
Since
s
n
>
0
for all
n
,
1
/s
n
<
1
/M
for all
n > N
implies
s
n
> M
for all
n > N
. Therefore,
s
n
→ ∞
.
Exercises 2.2
1. Prove or give a counterexample.
(a) If
s
n
→
s
and
s
n
>
0
for all
n
,
then
s >
0.
(b) If
{
s
n
}
and
{
t
n
}
are divergent sequences, then
{
s
n
+
t
n
}
is divergent.
(c) If
{
s
n
}
and
{
t
n
}
are divergent sequences, then
{
s
n
t
n
}
is divergent.
(d) If
{
s
n
}
and
{
s
n
+
t
n
}
are convergent sequences, then
{
t
n
}
is convergent.
(e) If
{
s
n
}
and
{
s
n
t
n
}
are convergent sequences, then
{
t
n
}
is convergent.
(f) If
{
s
n
}
is not bounded above, then
{
s
n
}
diverges to +
∞
.
2. Determine the convergence or divergence of{sn}. Find any limits that exist.17
(a)
s
n
=
3

2
n
1 +
n
(b)
s
n
=
(

1)
n
n
+ 2
(c)
s
n
=
(

1)
n
n
2
n

1
(d)
s
n
=
2
3
n
3
2
n
(e)
s
n
=
n
2

2
n
+ 1
(f)
s
n
=
1 +
n
+
n
2
1 + 3
n
3. Prove the following:
(a)
lim
n
→∞
n
2
+ 1

n
= 0.
(b)
lim
n
→∞
n
2
+
n

n
=
1
2
.
4. Prove Theorem 4.
5. Prove Theorem 6.
6. Let
{
s
n
}
,
{
t
n
}
,
and
{
u
n
}
be sequnces such that
s
n
≤
t
n
≤
u
n
for all
n
. Prove that if
s
n
→
L
and
u
n
→
L
,
then
t
n
→
L
.
II.3.
MONOTONE SEQUENCES AND CAUCHY SEQUENCES
Monotone Sequences
Definition 4.
A sequence
{
s
n
}
is
increasing
if
s
n
≤
s
n
+1
for all
n
;
{
s
n
}
is
decreasing
if
s
n
≥
s
n
+1
for all
n
. A sequence is
monotone
if it is increasing or if it is decreasing.
Examples
(a) 1
,
1
2
,
1
3
,
1
4
, . . .,
1
n
, . . .
is a decreasing sequence.
(b) 2
,
4
,
8
,
16
, . . .,
2
n
, . . .
is an increasing sequence.
(c) 1
,
1
,
3
,
3
,
5
,
5
, . . .,
2
n

1
,
2
n

1
, . . .
is an increasing sequence.
(d) 1
,
1
2
,
3
,
1
4
,
5
, . . .
is not monotonic.
Some methods for showing monotonicity:
(a) To show that a sequence is increasing, show that
s
n
+1
s
n
≥
1
for all
n
. For decreasing, show
s
n
+1
s
n
≤
1
for all
n
.
The sequence
s
n
=
n
n
+ 1
is increasing: Since
s
n
+1
s
n
=
(
n
+ 1)
/
(
n
+ 2)
n/
(
n
+ 1)
=
n
+ 1
n
+ 2
·
n
+ 1
n
=
n
2
+ 2
n
+ 1
n
2
+ 2
n
>
1
18
(b) By induction. For example, let
{
s
n
}
be the sequence defined recursively by
s
n
+1
= 1 +
√
s
n
,
s
1
= 1
.
We show that
{
s
n
}
is increasing. Let
S
be the set of positive integers for which
s
k
+1
≥
s
k
.
Since
s
2
= 1 +
√
1 = 2
>
1
,
1
∈
S
. Assume that
k
∈
S
;
that is, that
s
k
+1
≥
s
k
. Consider
s
k
+2
:
s
k
+2
= 1 +
√
s
k
+1
≥
1 +
√
s
k
=
s
k
+1
.
Therefore,
s
k
+1
∈
S
and
{
s
n
}
is increasing.
THEOREM 8.
A monotone sequence is convergent if and only if it is bounded.
Proof:
Let
{
s
n
}
be a monotone sequence.
If
{
s
n
}
is convergent, then it is bounded (Theorem 2).
Now suppose that
{
s
n
}
is a bounded, monotone sequence.
In particular, suppose
{
s
n
}
is
increasing. Let
u
= sup
{
s
n
}
and let
be a positive number. Then there exists a positive integer
N
such that
u

< s
N
≤
u
. Since
{
s
n
}
is increasing,
u

< s
n
≤
u
for all
n > N
. Therefore,

u

s
n

<
for all
n > N
and
s
n
→
u
.
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 Fall '08
 Staff
 Real Numbers, Integers, Limit of a sequence, Sn