The author introduces the "Maximal Principle" in order to prove that every vector space has a basis. For reference, I'll restate it as he wrote it: "Let F be a family of sets. If, for each chain C [subset of] F, there exists a member of F that contains each member of C, then F contains a maximal member". If a chain C of F is finite, then by comparing all the members of C, we should be able to find a member in C (hence in F) that contains all the others. Therefore, if the hypothesis are to fail anywhere, it should be possible for an infinite chain to not have a member that contains all the others. But in his proof that every vector space has a basis, the author states: "But since C is a chain, one of these [members of C], say A, contains all the others" without knowing whether C is finite. Could someone clear my head for me?