Convergent Sequence is Cauchy Sequence

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

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

Remark
A metric space in which the converse also holds is called complete.

An example of a complete metric space is given by the real number line.

Thus, every Cauchy sequence in $\R$ is convergent.

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

Let $\epsilon > 0$.

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

Because $\left \langle {x_n} \right \rangle$ converges to $l$, we have:
 * $\exists N: \forall n > N: d \left({x_n, l}\right) < \dfrac \epsilon 2$

In the same way:
 * $\forall m > N: d \left({x_m, l}\right) < \dfrac \epsilon 2$

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

Thus $\left \langle {x_n} \right \rangle$ is a Cauchy sequence.