ANALYSIS
REAL VARIABLE
SEQUENCES
A monotonic increasing sequence either converges to a finite limit or diverges
to =
∞
.
Proof
Let
S
be the set of numbers in the sequence
{
a
n
}
.
Case (i)
S
has no upper bound.
Given any
X,
∃
n
0
/ a
n
0
> X
⇒
a
n
> X
for
n
≥
n
0
.
This means that the sequence diverges to +
∞
.
Case (ii)
S
has an upper bound;
bda
n
=
a
.
Given
ε >
0
∃
n
1
/ a
n
1
> a

ε
⇒
a

ε < a
n
≤
a
for
n
≥
n
1
.
This means that the sequence converges to
a
.
Upper and lower limits
Definition
lim
n
→∞
sup a
n
lim
n
→∞
a
n
)
= lim
n
→∞
sup
m
≥
n
a
m
lim
n
→∞
inf a
n
lim
n
→∞
a
n
)
= lim
n
→∞
inf
m
≥
n
a
m
Justification (for lim sup) Case (i)
Suppose
S
has no upper bound.
i.e
sup
m
≥
1
a
m
= +
∞
Then
sup
m
≥
n
a
m
= +
∞
for all
n
.
In these circumstances we say that
lim
n
→∞
a
n
= +
∞
Case (ii)
Now suppose
S
has an upper bound.
Write
sup
m
≥
n
a
m
=
A
(
n
)
A
(1)
A
(2)
. . .
is a monotonic decreasing sequence. If this sequence is
bounded below, then it converges to a number Λ, and
lim
n
→∞
a
n
=
Λ, and if not,
lim
n
→∞
a
n
=
∞
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.
 Spring '98
 pfitz

Click to edit the document details