This is one of the exercise problems I found in Halmos' 'Finite Dimensional Vector Spaces':
Let $V$ be a finite-dimensional vector space and $M$ and $N$ be two of its subspaces. Then, if $$M \cup N = V$$ then either $M=V$ or $N=V$.
I'm trying a proof by contradiction. I assume the conclusion in false. Then neither $M$ nor $N$ can be of the dimension of $V$ because if it were, then it would be equal to $V$ (this I have proven). So both the subspaces will have a lower dimension than $V$. Now all I need to show is that the union of these two subspaces will miss out some element of $V$. I'm stuck at this point and would be very much obliged if someone could give me a hint. Thanks!