For example, let us consider two subspaces S & T of the vector space R3 where S = {(x,y,z) : y=0 , z=0} T = {(x,y,z) : x=0, z=0}
let a=(1,0,0) belongs to S & b=(0,1,0) belongs to T. But a+b does not belong to SUT
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 .