A
sequence {a_n} is a Cauchy Sequence if and only if for every E>0 there exists a positive integer N such that if m,n >= N then |a_n - a_m|<>
E (Read E as epsilon)
Basically a sequence, {a1,a2,a3,a4,a5...an}, is Cauchy if the terms in the sequence begin to 'get closer and closer together' as you get farther away from a1.
An example of a Cauchy Sequence would be:
{1/n}
The terms of this sequence are {1, 0.5, 0.33, 0.25, 0.2, 0.17...,.01 ,...1/n}
This sequence is Cauchy because as n goes to infinity the terms get closer and closer together. This process of 'getting closer and closer together' is also known as
convergence.