Show there exists $v\in V$ such that $v\not\in V_{i}$ for any $1\leq i\leq k$ 
Show that if $V$ is a nonzero vector space over $\mathbb{R}$, and $V_{1}, V_{2},\ldots V_{k}$ are proper subspaces of $V$ then there exists $v\in V$ such that $v\not\in V_{i}$ for any $1\leq i\leq k$

I can prove the case for $k=2$ but I cannot produce an induction argument. 

A proof using some topology in the finite dimensional case (in the general case and using only linear algebra, see learnmore's answer) : 
If each of the $V_i$ is proper then they're all closed sets with empty interior, which, according to the Baire Category theorem implies that their union also has empty interior : therefore it isn't $V$.
