Upper Bound is Dual to Lower Bound

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

Let $a \in S$ and $T \subseteq S$.

The following are dual statements:


 * $a$ is an upper bound for $T$
 * $a$ is a lower bound for $T$

Proof
By definition, $a$ is an upper bound for $T$ :


 * $\forall t \in T: t \preceq a$

The dual of this statement is:


 * $\forall t \in T: a \preceq t$

by Dual Pairs (Order Theory).

By definition, this means $a$ is a lower bound for $T$.

The converse follows from Dual of Dual Statement (Order Theory).

Also see

 * Duality Principle (Order Theory)