Completeness of the real numbers

Completeness is the property that every nonempty set of real numbers bounded above has a least upper bound in the real numbers. It distinguishes the real number system from the rational numbers.

Least upper bound: The smallest real number that is greater than or equal to every element of a given set. Completeness guarantees this bound exists for every nonempty real set bounded above.
Ordered field: A field equipped with a total order compatible with addition and multiplication. Completeness extends the ordered-field axioms with a guarantee against missing limits.
Intermediate value theorem: A theorem stating that a continuous function takes every intermediate value between two values it attains. Its proof depends on the real line having no gaps.
Completeness of the rational numbers: A false property asserting that every nonempty rational set bounded above has a rational least upper bound. The rationals fail this property, as shown by sets approaching an irrational boundary.
Monotone convergence theorem: A theorem stating that every bounded monotone sequence of real numbers converges. Its proof extracts a limit from the supremum of the sequence's terms.
Upper bound: An element greater than or equal to every member of a specified set. The completeness condition applies precisely to nonempty sets that have an upper bound.
Extreme value theorem: A theorem stating that a continuous function on a closed bounded interval attains a maximum and minimum. Completeness helps ensure that bounded continuous images contain their endpoint values.
Archimedean property: The property that for any positive real numbers, some integer multiple of one exceeds the other. It is distinct from completeness: ordered fields can be Archimedean yet incomplete.
Cauchy sequence: A sequence whose terms become arbitrarily close to one another as the indices grow. Completeness ensures every Cauchy sequence of real numbers converges to a real limit.
Rational number: A number expressible as a ratio of two integers with nonzero denominator. The rationals form an ordered field but fail the least-upper-bound property.

Linked from 11 pages