Orthocomplement is Closed Linear Subspace

Theorem
Let $H$ be a Hilbert space, and let $A \subseteq H$ be a subset.

Then the orthocomplement $A^\perp$ of $A$ is a closed linear subspace of $H$.

Proof
Let $\{ x_n \} \subset A^\perp$ be a convergent sequence and $x$ be its limit.

Then, by the definition of orthocomplement, for all $n \in \N, y \in A:\left \langle x_n, y \right \rangle = 0$.

Passing to limit, we have for all $y \in A : \left \langle x, y \right \rangle = 0$ by Inner Product is Continuous.

So $x \in A^\perp$.

This shows that $A^\perp$ is closed.