Cauchy's Convergence Criterion/Real Numbers

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

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

Sufficient Condition
The conditions are shown to be equivalent.

Hence the result.

Also see

 * Real Number Line is Banach Space
 * Cauchy's Convergence Criterion on Complex Numbers