Infimum of Empty Set is Greatest Element

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

Suppose that $\map \inf \O$, the infimum of the empty set, exists in $S$.

Then $\forall s \in S: s \preceq \map \inf \O$.

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


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

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


$\forall s \in S: s \preceq \map \inf \O$


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