I think you're making things a bit more complicated than they have to be.
Lets start with "if a sequence converges, then it is Cauchy," and forget the formal proof for a second. Intuitively, what this says is that if a sequence is getting closer and closer to a single point, then the points in the sequence must be getting closer and closer to each other. Makes sense, right?
As for the proof, since the sequence converges, we have that (sn) approaches s as n goes to infinity. Thus for any number (here, that number is epsilon/2), there is an N such that if n>N then . This is because approaches 0 as n-->infinity. The key point is that epsilon is arbitrarily chosen.
As for your question, simply find an N such that 1/N < ....and try go from there.