Greatest Lower Bound Property

Theorem
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 known as the greatest lower bound property of the real numbers.

Proof
Let $t = \set {x \in \R: -x \in S}$.

The idea is that $l$ is a lower bound of $S$ $-l$ is an upper bound of $T$.