User:Omlett71/Sandbox/Ordering on Real Numbers

Definition
The standard ordering on the real numbers is a total order $\le$ defined by a positivity property; that is, certain elements $x \in \R$ satisfy $P(x)$, and $y \le z \iff P(z-y) \lor y=z$. If the real numbers are characterized as equivalence classes of Cauchy sequences of rational numbers (see equivalence relation on cauchy sequences for more details), then the positive property can be defined in terms of the order on $\Q$ as below:

If $x$ is a Cauchy sequence in $\Q$, with $i$-th entry denoted $x_i$, then $P_\R([x]) \iff$ $\exists N \in \N$ such that $n \ge N \implies P_\Q(x_n)$. Informally, a real number is positive if a Cauchy sequence representing it is always positive after a certain entry.