---
tag: 0337CYB0VD
kind: definition
name: Order on the reals
msc: [26A03]
defines: 0MSSD1VCW9
uses: [0D8HE681SC, 09RFKMBQ63]
sources:
  - url: https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
    locator: Construction from Cauchy sequences
---

## Definition

Construct the real numbers as classes of Cauchy sequences of rational numbers, as in the [[construction from Cauchy sequences|0D8HE681SC]].
Let $x$ and $y$ be real numbers, represented by the Cauchy sequences $(x_n)$ and $(y_n)$ of rational numbers.
The notation $x < y$ means that there are a rational number $\varepsilon > 0$ and a natural number $N$ such that $y_n - x_n > \varepsilon$ for each $n \ge N$.
The notation $x \le y$ means that $x < y$ or $x = y$.

## Remarks

The relation $x < y$ does not depend on the choice of the sequences that represent $x$ and $y$.
For all real numbers $x$ and $y$, exactly one of $x < y$, $x = y$ and $y < x$ holds. Thus $\le$ is a total order.
On the rational numbers, this order agrees with the [[order of the rationals|09RFKMBQ63]].
With [[Dedekind cuts|04QNZZ4HHY]], the order is inclusion of the lower sets of the cuts.

## Explanations

### L1 human

Two real numbers are compared through their rational approximations.
Suppose that, from some point on, the approximations of $y$ exceed those of $x$ by a fixed positive gap.
Then $x$ is less than $y$.
For example, $1.41, 1.414, 1.4142, \dots$ approximates $\sqrt{2}$, and these approximations are larger than $1.4$ by more than $0.005$, thus $1.4 < \sqrt{2}$.