A problem in discrete geometry

[$\displaystyle \star$] Let $\displaystyle V$ be a finite dimensional vector space over some field $\displaystyle F$ and $\displaystyle \dim_F V=n.$ Let $\displaystyle \mathcal{A}$ be a finite set of the subspaces of $\displaystyle V$ and suppose that for any $\displaystyle \emptyset \neq \mathcal{B} \subseteq \mathcal{A}$ with $\displaystyle |\mathcal{B}| \leq n$ we have

$\displaystyle \bigcap_{W \in \mathcal{B}} W \neq \{0 \}.$ Prove that $\displaystyle \bigcap_{W \in \mathcal{A}} W \neq \{0 \}.$

[From now on I'll rate the problems that I give in here. It starts with one star, which means "fairly easy", and goes up to 5 stars, which means "unfairly difficult"! haha]