I have to show the following: Let $K_{1}, K_{2} \subseteq \mathbb{R}^{n}$ be convex cones with $K_{1} \cap K_{2} = \begin{Bmatrix} 0 \end{Bmatrix}$ and $intK_{i} \neq \varnothing , i=1,2$.
Show that $$\exists x \neq 0:x \in K_{1}^{*} \cap (-K_{2}^{*})$$, where $K_{i}^{*}$ is the polar set of $K_{i}$.
I tried a lot but without any results, which might be due to my lack of understanding of the polar set. I'd really appreciate if someone could show me how this works or give me a useful hint.
Thanks in advance and have a good day