2. Intervals in R

2.2. Archimedean property in R

Statement:

For every

x∈Rx \in ℝ

, there exists a natural number

n∈Nn \in ℕ

such that

n>xn > x

.
Equivalently,

Rℝ

R has no infinitely large or infinitesimal elements.

Consequences:

  • ∃n∈N\exists n \in ℕ

    such that

    1n<ϵ\frac{1}{n} < \epsilon

    for any

    ϵ>0\epsilon > 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.