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.