I have tried to research more about this following problem, and I have tried to come up with a proof myself, but unfortunately I can't think of anything that makes sense. I am having trouble with part (2) and (4) of this problem. Would really appreciate any help or clarification on how to begin solving this problem.
Question: Prove that every finitely generated vector space has a basis. In fact, every vector space has a basis, but the proof of that is beyond the scope of this course. Let V be a non-zero finitely generated vector space.
(1) Let w, v1,...,vn be in V . Prove w is in Span(v1,..., vn) if and only if Span(w, v1,..., vn) = Span(v1,..., vn).
(2) Prove, using part (1) that there exists a finite set X with nonzero members such that V = SpanX.
(3) Suppose V is spanned by a finite set of cardinality n. By part (2) we can assume 0 not in X. Our claim is V has a basis. We will use induction on n. Prove that the induction base holds. That is, if n = 1, then V has a basis.
(4) Induction hypothesis: suppose that V has a basis whenever it is spanned by a set of cardinality n. Now suppose V is spanned by a set of cardinality n + 1. Prove that V has a basis.