Archimedean Property

Key idea

For any real x there is an integer n > x.

In depth

For any real x there is a natural number n > x; equivalently, for ε > 0 there is n with 1/n < ε. It says there are no infinitely large or infinitely small real numbers, and follows from completeness.

Worked example

Given ε = 0.001, choose n = 1001, then 1/n < ε.

Real-world uses

• Showing that any small measurement can be repeatedly added to exceed a large target

Common mistake

Assuming it is automatic in all ordered fields; non-Archimedean fields exist.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

Expand all Collapse all

Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.
• Monotone Convergence Theorem – Bounded monotone sequences converge.