Infimum of Empty Set is Greatest Element

Theorem
Let $\left({S, \preceq}\right)$ be a poset.

Suppose that $\inf \varnothing$, the infimum of the empty set, exists.

Then $\forall s \in S: s \preceq \inf \varnothing$.

That is, $\inf \varnothing$ is the greatest element of $S$.

Proof
Observe that, vacuously, any $s \in S$ is a lower bound for $\varnothing$.

But for any lower bound $s$ of $\varnothing$, $s \preceq \inf \varnothing$ by definition of infimum.

Hence:
 * $\forall s \in S: s \preceq \inf \varnothing$

Also see

 * Supremum of Empty Set is Smallest Element