Least Upper Bound Property

Theorem
Let $$S \subset \R$$ be a non-empty subset of the set of real numbers such that $$S$$ is bounded above.

Then $$S$$ admits a supremum in $$\R$$.

This is sometimes called the least upper bound property of the real numbers.

Similarly, let $$S \subset \R$$ be a non-empty subset of the set of real numbers such that $$S$$ is bounded below.

Then $$S$$ admits an infimum in $$\R$$.

This is sometimes called the greatest lower bound property of the real numbers.

The two properties taken together are called the continuum property of $$\R$$, or the completeness axiom.

Proof
A direct consequence of Dedekind's Theorem.

Note
Not to be confused with:
 * The Continuum Hypothesis;
 * The Continuity Property.