Dedekind-Complete Bounded Ordered Set is Complete Lattice

Theorem
Let $\struct {L, \preceq}$ be an ordered set.

Let $L$ have a lower bound $\bot$ and an upper bound $\top$.

Let $\struct {L, \preceq}$ be Dedekind-complete.

Then $\struct {L, \preceq}$ is a complete lattice.

Proof
Let $S \subseteq L$.

If $S = \O$, then $S$ has a supremum of $\bot$ and an infimum of $\top$.

Let $S \ne \O$.

$S$ is bounded above by $\top$.

As $\struct {L, \preceq}$ is Dedekind complete, $S$ has a supremum.

$S$ is bounded below by $\bot$.

By Dedekind Completeness is Self-Dual, $S$ has an infimum.

Thus every subset of $L$ has a supremum and an infimum.

So, by definition, $\struct {L, \preceq}$ is a complete lattice.