Can you prove that if one subspace is contained in the other, the union is a subspace? That should be relatively straight-forward.
Yes, I think I get that part. Is it right like this?
If and are subspaces of and .
Then .
So is a subspace of
But how would you the other way. Assuming first that the union of two subspaces, say and , of the vector space . Then how do you prove that one of them is contained in the other?