How do you actually write out the terms in a Cauchy sequence?

Question

For example for $\epsilon>0$ there exist $N$ such $n,m>N$ implies $|s_m-s_n|<\epsilon$. I understand that intuitively, we don't need to know a certain limit and thus this definition for convergent sequences ends up being useful. But, I saw a proof that wrote out the terms like this: $|s_m-s_n|=|(s_m-s_{m-1})+(s_{m-1}-s_{m-2})+...+(s_{n-1}-s_n)|$ and I found it a bit strange since I assumed it was: $|(s_m-s_n)|=$(sequence of $s_m$)$-$(sequence of $s_n$). But upon writing this question, did they actually do the plus and minus trick? Such that it's only true because the insides get canceled out and you end up with $|s_m-s_n|$? Is my intuitive definition correct, in the least?

Answer 1

Yes. You can always do such manipulations. But your assumption is incorrect, check the proper definition of cauchy sequences find the error. What it implies is after a certain N, the following "terms" of the sequence get arbitrarily close to each other. the terms inside the modulus are m'th and n'th terms of the sequence for m,n>=N, some natural number depending upon choice of epsilon.