---
tag: 0A74SX55QP
kind: statement
role: proposition
name: Archimedean property
msc: [12J15]
uses: [0HQARMKEV5]
sources:
  - url: https://en.wikipedia.org/wiki/Archimedean_property
    locator: Ordered fields
same_as:
  wikidata: Q634579
  wikipedia: Archimedean property
---

## Statement

Let $F$ be an [[Archimedean ordered field|0HQARMKEV5]] and let $x$ be an element of $F$.
Then there is a natural number $n$ with $x < n$.

## Remarks

Proof: apply the definition with $y = 1$.
Thus the natural numbers are not [[bounded above|0B5H2V4JW8]] in $F$.
The [[real numbers are Archimedean|0A3GWZ86AY]], thus this holds for each real number $x$.

## Explanations

### L1 human

No element of an Archimedean field is larger than all the natural numbers.
For example, for the real number $x = 10^{6} + 0.5$, the natural number $n = 10^{6} + 1$ is larger.