Let us consider a set with an infinite number of vectors $\{v_k \mid k \in \mathbb{N} \}$ with $n$ cordinates, and if we consider the conned convex set containing all theses vectors, denoted by $K$.
My question is why if we take $a$ non zero vector, that does not belong to $K$, then we get that $$\langle a,v_k \rangle \leq 0,$$ for all $k \in \mathbb{N}$?