Least Upper Bound Property

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

Then $$S$$ admits a supremum in $$\mathbb{R}$$.

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

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

Then $$S$$ admits an infimum in $$\mathbb{R}$$.

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

Proof
A direct consequence of Dedekind's Theorem.