---
tag: 0D8HE681SC
kind: concept
name: Construction of the reals from Cauchy sequences
msc: [26A03]
sources:
  - url: https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
    locator: Construction from Cauchy sequences
---

## Summary

This construction defines the [[real numbers|08J2M1JYAG]] as limits of sequences of [[rational numbers|0NCJ5WXN6Z]].
A sequence $(x_n)$ of rational numbers is a Cauchy sequence when, for each rational $\varepsilon > 0$, there is $N$ with $|x_m - x_n| < \varepsilon$ for all $m, n \ge N$.
Two Cauchy sequences $(x_n)$ and $(y_n)$ are equivalent when $x_n - y_n$ tends to $0$: for each rational $\varepsilon > 0$, there is $N$ with $|x_n - y_n| < \varepsilon$ for all $n \ge N$.
A real number is an equivalence class of Cauchy sequences of rational numbers.
Addition and multiplication act term by term: $[(x_n)] + [(y_n)] = [(x_n + y_n)]$ and $[(x_n)] \cdot [(y_n)] = [(x_n y_n)]$.

## Remarks

The rational number $q$ corresponds to the class of the constant sequence $(q, q, q, \dots)$.
The sequence $1, 1.4, 1.41, 1.414, \dots$ of decimal approximations represents $\sqrt{2}$.
The result is a [[complete ordered field|0A9E6X0S3E]], with the [[order on the reals|0337CYB0VD]]. The same method completes any metric space.

## Explanations

### L1 human

A real number is something that rational numbers can approximate as well as you like.
The construction takes the approximating sequences themselves as the new numbers.
Two sequences that get closer and closer to each other count as the same real number.