Orthocomplement Reverses Subset

Theorem
Let $H$ be a Hilbert space.

Let $A, B$ be subsets of $H$, and let $A \subseteq B$.

Then $B^\perp \subseteq A^\perp$, where $\perp$ signifies orthocomplementation.

Proof
Suppose $h \in B^\perp$.

That is, for all $b \in B$: $h \perp b$.

As $A \subseteq B$, it follows that $h \perp a$ for all $a \in A$.

Hence, by definition of orthocomplement, $h \in A^\perp$.

Therefore, $B^\perp \subseteq A^\perp$.