Let $E$ be a non-empty subset of $\mathbb{R}^d$. The polar cone of $E$ is defined by
$$E^+ := \left\{p \in \mathbb{R}^d : p \cdot x \geq 0, \forall x \in E\right\}$$
and its bi-polar cone by
$$E^{++} = (E^+)^+ := \left\{\xi \in \mathbb{R}^d : \xi \cdot p \geq 0, \forall p \in E^+\right\}.$$
Show that $E^+$ and $E^{++}$ are closed convex cones and that $E \subset E^{++}$.