Reflexive Closure of Strict Ordering is Ordering
Let $S$ be a set.
Let $\prec$ be a strict ordering on $S$.
Let $\preceq$ be the reflexive closure of $\prec$.
Then $\preceq$ is an ordering.
Thus by Reflexive Closure of Transitive Antisymmetric Relation is Ordering, $\preceq$ is an ordering.