Cauchy Sequences of Real Numbers

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(a_{n})$ be a sequence in $\mathbb{R}$.

¶ $(a_{n})$ is a Cauchy sequence if for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $|a_{m}-a_{n}|<\varepsilon$ for all $m,n\in\mathbb{N}$ with $m,n\ge N$.