The problem statement, all variables and given/known data Let V be a vector space over an infinite field. Prove that V is not the union of finitely many proper subspaces of V. The attempt at a solution Suppose V is the union of the proper subspaces U1, ..., Un. Let ui be a vector not in Ui. If u1 + ... + un is in the union, then there must be some subspace that contains it. But then that subspace contains a sum where one of the terms doesn't belong to it. I'm hoping this isn't possible but I can't think of anything contradictory. In any case, I think my approach is wrong because I haven't really used the fact that V is defined over an infinite field and that the union is a finite one.