the first part is a quick result of Zorn's lemma: let be the set of all subspaces W of V such that since the set contains the 0 subspace and hence it's not empty.

choose any chain (totally ordered collection) of elements of then the union of those elements is still in thus has a maximal element by Zorn's lemma.

for the second part, suppose is a maximal element of and for some subspace of the claim is that : by maximality of in we must have

thus so we only need to show that suppose and let then and thus which

contradicts maximality of in