Cauchy's Convergence Criterion/Real Numbers

Theorem
Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Then $\left \langle {x_n} \right \rangle$ is convergent iff $\left \langle {x_n} \right \rangle$ is a Cauchy sequence.

Necessary Condition
Suppose $\left \langle {x_n} \right \rangle$ is convergent.

From Convergent Sequence is Cauchy Sequence, we have that every convergent sequence in a metric space is a Cauchy sequence.

We also have that the real number line is a metric space.

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

Sufficient Condition
Suppose $\left \langle {x_n} \right \rangle$ is a Cauchy sequence.

We have the result that a Cauchy Sequence Converges on Real Number Line.

Hence $\left \langle {x_n} \right \rangle$ is convergent.

The conditions have been shown to be equivalent.

Hence the result.