---
tag: 0A3GWZ86AY
kind: statement
role: theorem
name: The reals are Archimedean
msc: [26A03]
uses: [0QSPNASE50, 0HQARMKEV5, 0AXWET80YW]
sources:
  - url: https://en.wikipedia.org/wiki/Archimedean_property
    locator: Archimedean property of the real numbers
---

## Statement

The ordered field $\mathbb{R}$ of [[real numbers|08J2M1JYAG]] is an [[Archimedean ordered field|0HQARMKEV5]].
That is, for all real numbers $x$ and $y$ with $y > 0$, there is a natural number $n$ with $n y > x$.

## Remarks

The proof uses the [[completeness axiom|0AXWET80YW]]. If $\mathbb{N}$ were bounded above, let $b = \sup \mathbb{N}$. Then $b - 1$ is not an upper bound, so $n > b - 1$ for some $n$, and $n + 1 > b$, a contradiction.
Thus each [[complete ordered field|0A9E6X0S3E]] is Archimedean.

## Explanations

### L1 human

However small your step and however far your goal on the real line, finitely many steps get you past the goal.
There are no infinitely large real numbers.