Oh no, the N matters (and will vary depending on what sequence you are dealing with and what

you are given). For a Cauchy sequence, you have to be be able to pick a point for which the definition is true. The fact that you can do that is what makes the sequence Cauchy. You can't do that for the sequence

for example. It matters that no matter what N you pick, the terms after that don't get closer together. Successive terms are always 1 unit apart, and if you pick terms that are 1000 terms apart, the distance between them would not be

*small*, it would be a thousand--which, chances are, won't be smaller than any arbitrary

we are given. The fact that you

*can* choose some N and all the terms after it do what you say, that is, become really close together, is what makes the sequence Cauchy. And there is no

*difference* between a Cauchy sequence and a convergent one. Convergence implies Cauchy and Cauchy implies convergence. Saying a sequence is Cauchy is logically equivalent to saying it is convergent.