Cauchy's Convergence Criterion/Real Numbers

Theorem
Let $\sequence {x_n}$ be a sequence in $\R$.

Then $\sequence {x_n}$ is convergent $\sequence {x_n}$ is a Cauchy sequence.

Necessary Condition
Suppose $\sequence {x_n}$ is convergent.

From Convergent Sequence in Metric Space 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 $\sequence {x_n}$ is a Cauchy sequence.

Sufficient Condition
Suppose $\sequence {x_n}$ is a Cauchy sequence.

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

Hence $\sequence {x_n}$ is convergent.

The conditions have been shown to be equivalent.

Hence the result.

Also see

 * Cauchy's Convergence Criterion on Complex Numbers