The Order Axioms of ℝ Notes

2.2. Archimedean property in R

Statement:

For every

x∈R, there exists a natural number

n∈N such that

n>x.
Equivalently,

R has no infinitely large or infinitesimal elements.

Consequences:

• $\exists n \in ℕ$

  ∃n∈N such that

  $\frac{1}{n} < \epsilon$

  n1​<ϵ for any

  $\epsilon > 0$

  ϵ>0

• Ensures that ℕ is unbounded in ℝ

• Essential in constructing sequences that converge to zero

The Archimedean Property bridges the discrete world of natural numbers and the continuum of the real line.