$$\forall \epsilon > 0,\ \text{let} \ N=\frac{1}{\sqrt{\epsilon}}.$$$$\text{So}, \ m>n>N \implies \frac{1}{m^2}<\epsilon. $$$$|s_n-s_m| = \frac{n^2-1}{n^2}-\frac{m^2-1}{m^2}=\frac{1}{m^2}-\frac{1}{n^2} < \frac{1}{m^2} < \epsilon$$I am wondering about the definition of the Cauchy sequence and why most of the proofs I see do not assume WLOG $n>m$. In my recent assignment both my proofs immediately assumed $m>n$. Above is a shortened example. However, the proof presented in class and others I have seen online (for this problem and others) generally seem to avoid doing this. I'm pretty sure it's ok (since $|s_n-s_m|=|s_m-s_n|$) but I am wondering if there is some reason it not done more often.
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