Let be a field, let be a nonzero -vector space, and let be a linearly independent subset. Prove that there exists a basis for over which contains .
Let be the set of all linearly independent subsets of which contain . Note that is partially ordered by , and is nonempty since .
Let be a chain; that is, a linearly ordered subset. We claim that . First, certainly . To see that is linearly independent, let be a finite subset and suppose . Since is finite and is linearly ordered, there exists an element such that . Since is linearly independent, for all . Thus is linearly independent. So , and in fact is an upper bound for . By Zorn’s lemma, there exists a maximal element .
We claim that . To see this, suppose there exists an element . Suppose we have and (finitely many) such that . If , then , a contradiction. Thus , and we have . Since is linearly independent, for all . Thus is linearly independent and contains , violating the maximalness of in . Thus we have .
So is a linearly independent generating set (i.e. basis) for which contains .