Hey guys, I'm having trouble wrapping my head around Cauchy Sequences.

So, I know that the definition of such a sequence is if $\displaystyle m,n>N$ then $\displaystyle |a_n-a_m|<\epsilon$. This means that the sequence is getting epsilon close to each other far down the number line.

How do we know what to choose for $\displaystyle a_m$ and $\displaystyle a_n$ if we say $\displaystyle n>m>N$ when trying to prove that the sequence $\displaystyle X_n=1/n$ is Cauchy?

Also, the proof "if a sequence converges, then it is Cauchy," (bottom of first page here:

http://legacy.lclark.edu/~istavrov/advcalc-sept30-cauchy.pdf)

why is it true that $\displaystyle |s_n-s|<\epsilon /2$? Why is it less than epsilon over two and not just epsilon. I assume it has something to do with the triangle inequality?

If theres any trick to understanding this stuff please share!

Thanks!