Consider the sequence . It is easy to show, using induction that for all .
Hence, for all large . Hence is not bounded above.
Motivated by our remarks at the end of section 1.4, we have the following.
2.3.3
Theorem:
If is convergent then it is bounded
2.3.4
Example:
Consider the sequence . We showed in example 2.2.2(iii) that this sequence in not convergent. Clearly, it is a bounded sequence as for every n .
However, it not always easy to guess whether a sequence is convergent or not and even if it is convergent, what is its limit. We describe in next section in next section some theorems which helps us to compute limits of sequences.
. It is easy to show, using induction that 
for all 
. 
for all large 
. Hence 
is not bounded above. 
is convergent then it is bounded 
. We showed in example 2.2.2(iii) that this sequence in not convergent. Clearly, it is a bounded sequence as 
for every n . 
