Greatest Lower Bound Theorem

Greatest Lower Bound Theorem: If $E\subseteq \mathbb{R}$ is nonempty and bounded below , then $\inf E$ exists in $\mathbb{R}$.

This is the “lower” counterpart to the least upper bound theorem and follows immediately by applying the supremum property to $-E=\{-x:x\in E\}$.

Proof sketch (optional): If $E$ is bounded below, then $-E$ is bounded above. Let $s=\sup(-E)$. Then $-s$ is the greatest lower bound of $E$, i.e., $\inf E=-s$.