## 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: Suppose converges to Choose then and there is a positive integer such that implies and hence by the triangle inequality hence is 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 &gt;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  