Show that if the union of a finite number of vector subspaces $F_1, \ldots, F_r$ of $E$ is a vector subspace, then one of the vector subspaces $F_i$ contains all the others.
Show that if the union of a finite number of vector subspaces $F_1, \ldots, F_r$ of $E$ is a vector subspace, then one of the vector subspaces $F_i$ contains all the others.