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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 [...].