V is any vector space, where are subspace of V.
Prove U is not a subspace of the vector space V.
You can't prove this, It isn't true unless you have restrictions on and . For example, if the vector space is , , and then which certainly is a subspace.
What is true is that is a subspace if and only if one is a subspace of the other.
In order to prove this, you will have to make use of the fact, which isn't true unless one is NOT a subspace of the other, that there exist a vector u in that is not in and that there exist a vector v in that is not in .