Q:

Prove a sequence converges using the definition of a limit? 

Here is the sequence:
$$a_n = \frac{n^2}{cn^2 + 1} \mbox{ where } c  < 0.$$
If I prove this function has a limit using the limit definition, as $n$ goes to infinity, does that prove the sequence converges? 

A:

Yes If you are able to find a real number to which the sequence approaches as n tends to infinity the sequence then you can say that the sequence converges. 
I think,you can see directly this from the definition of a "Convergent Sequence" 
Read this page I think it will clear all your doubts http://en.wikipedia.org/wiki/Limit_of_a_sequence

