Cauchy Sequences

Supposeconverges 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.


A sequenceis Cauchy if and only if for eachthere is a positive integersuch that ifthen

Theorem Every convergent sequence is a Cauchy sequence.

Proof: Supposeconverges toChoosethenand there is a positive integersuch thatimpliesandhence 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: Provedoes not converge.

so take thenifso the sequence is not Cauchy and does not converge.

Example: Proveconverges.

Letthenso takethenando 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 intervalIf the term with largest magnitude outside this interval isthen all terms in the sequence lie in the interval