Convergent Sequence is Cauchy Sequence/Metric Space

Theorem
Let $M = \struct {A, d}$ be a metric space.

Every convergent sequence in $M$ is a Cauchy sequence.

Proof
Let $\sequence {x_n}$ be a sequence in $A$ that converges to the limit $l \in A$.

Let $\epsilon > 0$.

Then also $\dfrac \epsilon 2 > 0$.

Because $\sequence {x_n}$ converges to $l$, we have:
 * $\exists N: \forall n > N: \map d {x_n, l} < \dfrac \epsilon 2$

So if $m > N$ and $n > N$, then:

Thus $\sequence {x_n}$ is a Cauchy sequence.

Also see

 * Definition:Complete Metric Space, where the converse is true.
 * Real Convergent Sequence is Cauchy Sequence
 * Complex Sequence is Cauchy iff Convergent