Module 1
:
Real Numbers, Functions and Sequences
Lecture
3
:
Monotone Sequence and Limit theorem [ Section 3.2 : Limit Theorems on Sequences ]
3.2.1
Theorems (Algebra of limits):
Let
,
be sequences such that
and
.
(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
.
Proof
(i)
Let
be given. Choose
such that
Let .Then
.
This proves (i).
(ii)
To prove (ii), first note that
and
being convergent, are bounded sequences by theorem 1.5.3.
Back
Next
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
, 
be sequences such that 
and 
.
is convergent and 
is co
. 
is defined for all 
, for some 
and the sequence 
is convergent with
be given. Choose 
such that _clip_image002_0002.gif){kind=small}
_clip_image002_0001.gif){kind=small}
.
and 
being convergent, are bounded sequences by theorem 1.5.3. 
Next
