The major definition of Cauchy sequence is as below.
The sequense $\{ a_n\}\subset \mathbb{R}$ is a Cauchy sequence.
$\Leftrightarrow$
For all $\epsilon >0$, there exists $N\in \mathbb{N}$ s.t. $n,m>N \Rightarrow |a_n-a_m|<\epsilon.$
But is it O.K. that I suppose the magnitude relation of $n$ and $m$ ?
That is, is the definition below O.K ?
The sequense $\{ a_n\}\subset \mathbb{R}$ is a Cauchy sequence.
$\Leftrightarrow$
For all $\epsilon >0$, there exists $N\in \mathbb{N}$ s.t. $n>m>N \Rightarrow |a_n-a_m|<\epsilon.$
If this is O.K., I think the latter is more useful than the former because the latter has the magnitude relation of $n,m$ and that is easy to handle.
Best Answer
You can suppose that $n > m$ WLOG because the set of natural numbers is totally ordered.