Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Let $p \in \mathbb{R}^d$, set
$$K_p := \{x \in K : p \cdot x \leqslant p \cdot y, \forall y \in K\}.$$
Show that $K_p$ is non-empty, convex and closed and that $\operatorname{Ext}(K_p) \subset \operatorname{Ext}(K)$.