Definition:
Problem:
I understand $\sup_{x_1\in C_1} a'x_1 \le \inf_{x_2\in C_2} a'x_2$ and $\inf_{x_1\in C_1} a'x_1 < \sup_{x_2\in C_2} a'x_2$ will imply proper separation (first inequality indicates existence of separation hyperplane, second inequality indicates the hyperplan does not contain both $C_1,C_2$), but I have trouble understanding the converse, why proper separation implies the second inequality? Anyone can help show a proof?
Related background:
The first inequality is saying there exists a hyperplane that separates $C_1$ and $C_2$, as indicated by the book earlier,
(Taking into account the first inequality ) Proper means $ a'x_1 < a'x_2$ for at least one $x_1 \in C_1$ and one $x_2 \in C_2$, then second inequality follows trivially.