Let W be a subspace of V smaller than L. i.e. there is a vector 'x' in L [i.e. x is a linear combination of (v1,.., vn)] which is not in W
This leads to a contradiction under two facts about W:
1. W is a sub-space
2. W contains (v1,.., vn)
Under this contradiction - we can say any subspace with (v1,v2,,,vn) will have L as a subspace.