Complete Lattice has Both Greatest Element and Smallest Element

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Theorem

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

Then:

$(\text{1}) \quad \struct{S, \preceq}$ has a smallest element, namely:

$\quad \bot := \sup \O$

$(\text{2}) \quad \struct{S, \preceq}$ has a greatest element, namely:

$\quad \top := \inf \O$

Proof

From Complete Lattice is Bounded:

$\struct{S, \vee, \wedge, \preceq}$ is a bounded lattice

From Bounded Lattice has Both Greatest Element and Smallest Element:

$(\text{1}) \quad \struct{S, \preceq}$ has a smallest element, namely:

$\quad \bot := \sup \O$

$(\text{2}) \quad \struct{S, \preceq}$ has a greatest element, namely:

$\quad \top := \inf \O$

$\blacksquare$