Complete Lattice is Bounded

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

Then
 * $L$ is bounded

Proof
By definition of complete lattice:
 * $\O$ admits a supremum and an infimum.

By Infimum of Empty Set is Greatest Element:
 * $\forall x \in S: x \preceq \inf \O$

Thus by definition:
 * $L$ is bounded above.

By Supremum of Empty Set is Smallest Element:
 * $\forall x \in S: \sup \O \preceq x$

Thus by definition:
 * $L$ is bounded below.

Hence $L$ is bounded.