Cauchy Sequences
Suppose
converges to a real number A. The terms of the sequence must get close to A. If a-n and a-m are both close to A then they are close to each other.
Definition
A sequence
is Cauchy if and only if for each
there is a positive integer
such that if
then
Theorem Every convergent sequence is a Cauchy sequence.
Proof: Supposeconverges to
Choosethen
and there is a positive integer
such that
implies
andhence by the triangle inequality 
henceis Cauchy.
If a sequence is convergent, it must be Cauchy. If it is not Cauchy, it is not convergent. To prove a sequence converges, it is enough to prove it is Cauchy. To prove it does not converge, it is enough to prove it is not Cauchy.
Example: Prove
does not converge.
so take
then
if
so the sequence is not Cauchy and does not converge.
Example: Prove
converges.
Let
then
so take
then
and
o the sequence is Cauchy and converges.
Every Cauchy sequence is bounded. This is because if the sequence converges to A, for %epsilon >0 all but a finite number of terms in the sequence lie inside the interval
If the term with largest magnitude outside this interval is
then all terms in the sequence lie in the interval
