Show that if U is a subspace of a vector space V that is in turn a subspace of a vector space W, then U is a subspace of W
If U is a subset of W, and it is a vector space over the same field with addition and scalar multiplication identical with the restriction of the equivalent operations on W to U then it is a subspace of W. That U is a subspace of V a subspace of W is sufficient to justify this last requirement.
CB