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 $$\frac \epsilon 2 > 0$$.

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

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

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

$$ $$ $$

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