Least Upper Bound Theorem - Imported Knowlpedia Content

Least Upper Bound Theorem: If $E\subseteq \mathbb{R}$ is nonempty and bounded above, then $\sup E$ exists in $\mathbb{R}$ .

Remarks

This theorem is the working form of completeness: it guarantees the existence of optimal bounds and is used to prove convergence of monotone sequences, the existence of limits, and many properties of continuous functions.