Module 1
:
Real Numbers, Functions and Sequences
Lecture
2
:
Convergent & Bounded Sequences [ Section 2.3 : Bounded Sequences ]
2.3
Bounded Sequences
We saw that sequence
is not convergent as its terms keep growing bigger and bigger. This motivates our next definition.
2.3.1
Definition:
(i)
Let
be a sequence.
We say
is
bounded above
if there is some real number
such that
.
Sequence bounded above by
(ii)
We say
is
bounded below
if there exists a real number
such that
.
Sequence bounded below by
(iii)
We say
is
bounded
if it is both bounded above and below.
Bounded Sequence
2.3.2
Examples:
(i)
Consider the sequence
. Since for every
,
, the sequence is bounded.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
is not convergent as its terms keep growing bigger and bigger. This motivates our next definition. 
be a sequence. 
such that 
. 
such that 
. 
. Since for every 
, 
, the sequence is bounded. 
