On Cauchy Sequences

I would consider this a soft question because I am seeking some insight on how to work with Cauchy sequences by using the Cauchy criterion for convergence.

To my understanding, the definition is given $\epsilon > 0$, there are sequences $a_n$ and $a_m > N$ such that, for $m>n >>N$, $|a_m - a_n| \approx_{\epsilon} N$. I have seen that my professor usually takes the differences between successive terms and forms a telescoping series in order to find the convergence. I am curious if there are other ways to approach such a problem without using a telescoping series. That is, are there other ways to find the convergence of a Cauchy sequence, and if so, what are they?

Thanks in advance for any input in this discussion.

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Just to clarify, there is only one sequence in the definition. Let the sequence be $\{a_i\}_{i \in \mathbb{N}}$. The definition is: for all $\epsilon > 0$, there is a $N \in \mathbb{N}$ such that $n, m > N$ implies $|a_n - a_m| < \epsilon$. – Pratyush Sarkar Oct 17 '13 at 4:07
Also, if you are working on $\mathbb{R}$, then Cauchy sequences are equivalent to convergent sequences. – Pratyush Sarkar Oct 17 '13 at 4:09
Yes, working in $\mathbb{R}$, sorry about that. – Jamil_V Oct 17 '13 at 4:11
@ Jamil: If you want to show that any cauchy sequence in $\mathbb R$ converges, then you can apply nested interval theorem to prove that. Also, this is a standard result as $\mathbb R$ is complete. – Anupam Oct 17 '13 at 4:11
@ Jamil: youtube.com/watch?v=F5gVTCm-P14 – Anupam Oct 17 '13 at 4:14