Equivalence of Definitions of Order Complete Set

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

Definition 1 implies Definition 2
Let $\struct {S, \preceq}$ be an order complete set by definition 1.

Let $H \subseteq S$ have a lower bound.

Let $K$ be the set of all lower bounds of $H$.

Then $K \ne \O$.

By definition of lower bound:
 * $\forall x \in K: \forall y \in H: x \le y$

and so all elements of $H$ are upper bounds of $K$.

Thus by hypothesis $K$ admits a supremum.

Let $k = \map \sup K$.

By definition of supremum, $k$ precedes every upper bound of $K$.

In particular, $k$ precedes every element of $H$.

Thus $k$ is a lower bound of $H$.

But we have that $k$ is an upper bound of $K$.

That is, $k$ succeeds every lower bound of $H$.

That is, $k$ is an infimum of $H$.

Thus $\struct {S, \preceq}$ is an order complete set by definition 2.

Definition 2 implies Definition 1
Let $\struct {S, \preceq}$ be an order complete set by definition 2.

Let $H \subseteq S$ have an upper bound.

Let $K$ be the set of all upper bounds of $H$.

Then $K \ne \O$.

By definition of upper bound:
 * $\forall x \in K: \forall y \in H: y \le x$

and so all elements of $H$ are lower bounds of $K$.

Thus by hypothesis $K$ admits an infimum.

Let $k = \map \inf K$.

By definition of infimum, $k$ succeeds every lower bound of $K$.

In particular, $k$ succeeds every element of $H$.

Thus $k$ is an upper bound of $H$.

But we have that $k$ is a lower bound of $K$.

That is, $k$ precedes every upper bound of $H$.

That is, $k$ is a supremum of $H$.

Thus $\struct {S, \preceq}$ is an order complete set by definition 1.