Math and science::Analysis::Tao::05. The real numbers

# Archimedean property

The Archimedean property is a property of the reals. Roughly, it says: there are no infinitely small or infinitely large elements, when compared to the rationals. It also says that the naturals are not bounded by any real.

### Archimedean property

1. Given any number $$x \in \mathbb{R}$$, there exists an $$n \in \mathbb{N}$$ satisfying [...].
2. Given any real number $$x > 0$$, there exists an $$n \in \mathbb{N}$$ satisfying [...].

This is Abbott's description of the Archimedean property.

Tao presents the more common form which goes like:

For any two positive reals $$x$$ and $$y$$ there exists a natural number $$n$$ such that [...].

### Density of $$\mathbb{Q}$$ in $$\mathbb{R}$$

The Archimedean property leads to an important result:

For any reals $$a, b \in \mathbb{R}$$ such that $$a < b$$ there exists a rational $$q \in \mathbb{Q}$$ such that [...].