With a general sequence we usually have to say what the set
is from which the elements of the sequence have been chosen. Usually
we are dealing with one of the usual classes of number, namely the
rational numbers, the real numbers, or the complex numbers, but the
definition of Cauchy Sequence works in any setting where we have a
concept of "distance". 

A "Cauchy Sequence" is a general sequence with the following property:
 You think of a distance, then
 I say where to start, then
 you find that from then on,
 any two elements in the sequence differ by no more than your stated distance.
That sounds convoluted, but the idea is that the sequence can bounce
around as much as it likes to start with, but after some point it
settles down and all the elements are close. Indeed, you say how close
you want them to be, and we find that if we go far enough, everything
is within that tolerance.
Equivalence classes of Cauchy sequences of rational numbers are one
way to construct the real numbers.
Local neighbourhood  D3