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.
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