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.