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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?

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One would expect. Of course, you should have a candidate L in mind for the value of the limit. Then all you need to do is show that $a_{n} - L$ can be made arbitrarily small in absolute value by choosing $n$ large enough. – Chris Leary Oct 24 '11 at 17:21

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

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