Def/Least upper bound

 =Uniqueness=

The use of the article the least upper bound is justified by its uniqueness (when it exists). Namely, if $S$ is a subset of a poset $(X, \leq)$, and $x_1, x_2$ are two least upper bounds for $S$, we find that $x_1 \leq x_2$ and $x_2 \leq x_1$. It follows, from anti-symmetry that $x_1 = x_2$.

Hence, if a subset $S$ has a least upper bound, then it has a unique least upper bound. Hence we call it the least upper bound.

=Metadata=