# Least upper bound

Given a subset $S$ in some larger partially ordered set $R$, a least upper bound or supremum, for $S$ is an element $M \in R$ such that $s \leq M$ for every $s \in S$ and there is no $m < M$ with this same property.

If the least upper bound $M$ of $S$ is an element of $S$, it is also the maximum of $S$. If $M \not\in S$, then $S$ has no maximum.

Completeness: This is one of the fundamental axioms of real analysis.

A set $S$ is said to be complete if any nonempty subset of $S$ that is bounded above has a supremum.

The fact that $\mathbb{R}$ is complete is something intuitively clear but impossible to prove using only the field and order properties of $\mathbb{R}$