Least Upper Bound Theorem

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

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.

Proof sketch (optional): This is typically taken as an axiom (completeness) or proved from an equivalent completeness formulation (e.g., Cauchy completeness or nested intervals). The main idea is that the “gap-free” nature of $\mathbb{R}$ forces a least upper bound to exist.