Question: Let be subspaces of vector space . Prove that is a subspace of if and only if or .

I managed to prove it one way. I managed to show that if or , then is a subspace of .

However, I cannot go the other way. Surely the zero vector is contained in because it is contained in and thus their union. So that's not the problem; the problem seems to be whether or not the structure is closed under addition.

So I imagine that if neither nor are subsets of each other, then there exists such that and there exists such that . And somehow, I need to show that there exists such that from that. This would show that if neither nor hold, then fails to be a subspace of

How do I do so? Am I even thinking on the right track?

Thanks!