Module 1
:
Real Numbers, Functions and Sequences
Lecture
3
:
Monotone Sequence and Limit theorem [ Section 3.2 : Limit Theorems on Sequences ]
3.2
Limit Theorems on sequences :
Some more theorems which helps us in computing limits of sequences are as follows:
3.2.1
Theorems (Algebra of limits):
Let
,
be sequences such that
and
. Then the following hold:
(I)
A sequence
is convergent and
.
(ii)
A sequence
is co
nvergen
t and
.
(iii)
If
is defined for all
, for some
and the sequence
is convergent with
.
3.2.2
Sandwich Theorem:
Let
for all
. If
and
, then
.
3.2.3
Examples (Some important limits):
(i)
Let
with
. Then,
as
. To see this, note that for
we can write
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
, 
be sequences such that 
and 
. Then the following hold: 
is convergent and 
is co
. 
is defined for all 
, for some 
and the sequence 
is convergent with
for all 
. If 
and 
, then 
. 
with 
. Then, 
as 
. To see this, note that for 
we can write
