Hi,
Suppose that V is a vector space and is a nonempty collection of subspaces of V.
I want to show that if is a chain, then is a subspace of V.
Intuitively, this is obvious but I don't know how to prove that is closed under union of chains.
Then I think I could use Zorn's lemma to say that it has a maximal element that is a subspace of V.