$W$ is a subspace of $\mathbb{R}^n$ and $K$ is a compact subset of $V$ with $W \cap K = \emptyset$. Suppose $W$ is a subspace of $\mathbb{R}^n$ and $K$ is a compact subset of $V$ with $W \cap K = \emptyset$. Show that there exists a vector $v \in V$ such that $\langle v,w \rangle = 0$ for all $w\in W$ and $\langle w,x \rangle <0$ for all $x\in K$.
There is also a hint for this problem: Define $A=\{k-w : k\in K, w\in W \}$ and use Farkas Lemma. 
I don't know how to apply the hint or how the fact that K is compact influences anything related to the Farkas Lemma.      
 A: Here is my try, please someone check whether its correct or not.
We need the following lemma: 
Let $K \in \mathbb{R}^n$ be a closed and convex set with $ 0 \not\in K$ then there exists a vector $V \in \mathbb{R}^n$ such that $\langle v,x \rangle >0$ for every $x\in K$.
And the geometric interpretation of the Farkas' Lemma, which is here : http://en.wikipedia.org/wiki/Farkas%27_lemma
Solution:
As by the hint we define $A = \{k-w : k \in K,w \in W \}$. Since $W \cap K = \emptyset$ then it is clear that $0 \not\in A$ so by the lemma there exist a $v \in \mathbb{R}^n$ such that $\langle v,x \rangle >0$ for every $x \in A$, which implies that there exists a $v \in \mathbb{R}^n$ such that $\langle v,k-w \rangle >0$ for every $k \in K$ and $w \in W$. So we have that :
$$\langle v,k-w \rangle >0$$
$$\langle v,k\rangle >\langle v,w \rangle $$ , so 
and since $K$ is bounded while $W$ is not bounded then the $RHS$ is bounded while the $LHS$ can go to infinity, so we must that that $LHS = 0$,so $\langle v,w \rangle=0 $ (Does this part maybe need some more explanation?).
Now since $W$ is a subspace and $W \cap K = \emptyset$ we cant find any $\alpha_1, \cdots, \alpha_m \in \mathbb{R}$ such that $x = \alpha_1 w_1 + \cdots \alpha_n w_n$ where $x\in K$ and $w_1,\cdots w_n$ are a basis of $W$. Then by Farkas' Lemma we have that there exists a $v \in \mathbb{R}^n$ such that $\langle v,x \rangle <0$ for every $x\in K$ (the second scenario will happen since the first cant happen. This concludes our proof.
Does this look correct, and is there anything unclear or that needs a proof?
