Bottom in Compact Subset

Theorem
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

Then $\bot \in K\left({L}\right)$

where $\bot$ denotes the smallest element of $L$,
 * $K\left({L}\right)$ denotes the compact subset of $L$.

Proof
By Bottom is Way Below Any Element:
 * $\bot \ll \bot$

where $\ll$ is the way below relation.

By definition:
 * $\bot$ is compact.

Thus by definition of compact subset:
 * $\bot \in K\left({L}\right)$