I am reading materials about convex optimizations. It says: If $Z$ is a closed convex cone, then $Z$ is the anti-dual of its anti-dual cone $Z_-$, i.e.,
$$ Z=\{z\in \mathbf{R}^{L+1}:z^T\zeta\leq0,\forall\zeta\in Z_-\}, $$ $$ Z_-=\{\zeta\in\mathbf{R}^{L+1}:\zeta^Tz\leq0,\forall z\in Z\}. $$
How can I prove this? Why is this property useful?