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.

Sufficient 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.

Necessary 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.