Section 2.2:
Algebraic and Order Limit Theorems
Consider sequences
(
)
n
a
and
( )
n
b
. From these, we construct the following sequences, the sum sequence,
(
)
n
n
a
b
+
, the product sequence,
(
)
n n
a b
, and the quotient sequence
(
29
n
n
a
b
. In this section, we consider the natural
question, “if
and , do
(
)
n
n
a
b
+
,
(
)
n n
a b
, and
(
29
n
n
a
b
converge to
A
B
+
,
AB
, and
A
B
, respectively?”
As we might suspect, sequences behave quite nicely under normal arithmetic operations of addition and
multiplication. Although it is also true for division, the proof is a bit trickier. Let us deal with the sequence
formed by the sum first as it is the easiest to deal with.
Theorem:
Suppose
(
)
n
a
and
( )
n
b
are sequences such that
and . Then the sum sequence
(
)
n
n
a
b
+
converges to
A
B
+
Proof
Consider
0
ε
. Since , there is some
1
N
such that for all
1
n
N
≥
,
/ 2
n
a
A

<
. Similarly, there is
2
N
such that for all
2
n
N
≥
,
/ 2
n
b
B

<
. Therefore, pick
{
}
1
2
max
,
N
N N
=
. It follows that for all
n
N
≥
,
n
n
n
n
n
n
a
b
A
B
a
A b
B
a
A
b
B
+


=

+

≤

+

<
Hence .
Note that in the proof above, after we fixed the value for
, we were able to force
/ 2
n
a
A

<
for
n
large
enough. This might seem insignificant at first, but consider why we were able to do this. The
convergence
of
(
)
n
a
is quite powerful as it allows us to make the term
n
a
A

as small as we want. You could think of it as
having “control” over this term. Also note that the
triangle inequality
is used in the proof above. As stated
earlier, this inequality will be
extremely
useful throughout our study of analysis.
Our next question deals with the product of two convergent sequences. We claim that
(
)
n n
a b
converges to
AB
provided that
(
)
n
a
converges to
A
and
( )
n
b
converges to
B
.
Now, before we proceed to the formal proof, let us
consider how the product sequence behaves. In the end, what we wish to show is that for any given
0
, we
can make
n n
a b
AB

<
for all
n
sufficiently large. The following equality (another nice trick) will be helpful: