Cauchy Sequence

Let be a Metric space with distance function We say that a sequence in is Cauchy if for every there is some such that for all :

Lemma

Every convergent sequence is Cauchy.

Lemma

Every Cauchy sequence is Bounded.

Theorem

A real sequence is convergent if and only if is Cauchy. This easily extends to sequences in

Proof

We only need to show one direction. Assume is Cauchy. Because is Bounded, we can use Boltzano-Weierstrass theorem to find some and that converges. But then by the Cauchy condition, we can find that converges.