If U, U ′ are subspaces of V , then the union U ∪ U ′ is almost never a subspace (unless one happens to be contained in the other). Prove that, if W is a subspace, and U ∪ U ′ ⊂ W , then U + U ′ ⊂ W .
This seems fairly simple, but I am stuck on how to go about proving it.