Definition
Completeness of the Real Numbers
Every non-empty subset of the real numbers
that is bounded above has a supremum in . Similarly, every non-empty subset of
that is bounded below has an infimum in . This property distinguish the real numbers from the rational numbers
, which are “incomplete” (e.g., the set has no supremum in ).
Equivalent Formulations
The completeness of
- Cauchy Completeness: Every Cauchy sequence of real numbers is a convergent sequence.
- Nested Interval Property: For every sequence of nested closed intervals
whose lengths approach zero, there is exactly one point contained in all intervals. - Bolzano–Weierstrass Theorem: Every bounded sequence has a convergent subsequence.
- Monotone Convergence Theorem: Every bounded monotone sequence converges.