Let be non-empty subspaces of V. Prove that if is a subspace of V, than either (I don't know how to do this in Latex) or . ( symbol should mean proper subset here).
This should be done by contradiction I think. So assume that there exists and , wuch that .
Still assume now that is a subspace of V.
Now I'm having a brain fade. Hint perhaps?
Suppose, and then there's a vector, call it
We have so if the union is a subspace we should get
However and since would imply ( by definition of subspace, summing ) that analog. with
So our assumption and , must have been wrong
If one is contained in the other, the union is obviously a subspace, and so we are done