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

So, I know that the definition of such a sequence is if

then

. 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

and

if we say

when trying to prove that the sequence

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

? 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!