# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 97: The Archimedean Property of the Reals

$\mathbb{R}$ has the Archimedean property, which states that for any positive $x, y \in \mathbb{R}$ there exists an $n \in \mathbb{N}$ such that

$nx > y$.

Intuitively, this means that $\mathbb{R}$ contains neither infinitely large nor infinitesimally small numbers. (The property would not hold if $y$ was infinite, or if $x$ was infinitesimal.)

#### Proof:

Finding an $n$ such that $nx > y$ is equivalent to finding one such that $n > \frac{y}{x}$. If there is no such $n$, this means that $n \leq \frac{y}{x}$ for all $n$ and thus the number $\frac{y}{x}$ is an upper bound on $\mathbb{N}$. However, $\mathbb{N}$ has no upper bound, so this is a contradiction, meaning such an $n$ must exist.