Least upper bound

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Given a subset $S$ in some larger partially ordered set $R$, a least upper bound or supremum, for $S$ is an element $\displaystyle M \in R$ 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.


The Least Upper Bound Axiom: This is one of the fundamental axioms of real analysis. According to it, any nonempty set of real numbers that is bounded above has a supremum. This is something intuitively clear but impossible to prove using only the field properties, order properties and completeness property of the set of real numbers.