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Concept
Completeness of the real numbers
Summary
Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value. In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number.
Depending on the construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom), or may be a theorem proven from the construction. There are many equivalent forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness (completeness as a metric space).
The real numbers can be defined synthetically as an ordered field satisfying some version of the completeness axiom. Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are non Archimedean fields that are ordered and Cauchy complete. When the real numbers are instead constructed using a model, completeness becomes a theorem or collection of theorems.
Least-upper-bound property
The least-upper-bound property states that every nonempty subset of real numbers having an upper bound must have a least upper bound (or supremum) in the set of real numbers.
The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers
This set has an upper bound. However, this set has no least upper bound in Q: the least upper bound as a subset of the reals would be √2, but it does not exist in Q.
For any upper bound x ∈ Q, there is another upper bound y ∈ Q with y < x.
For instance, take x = 1.5, then x is certainly an upper bound of S, since x is positive and x^2 = 2.25 ≥ 2; that is, no element of S is larger than x.
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