Let p,r,q be sets. Prove that if p is a subset of r and r is a proper subset of q, then p is a proper subset of q
But you still need to prove that p is a proper subset of q! That is, that there exist a least one member of q that is not a member of p. To do that use the fact that r is a proper subset of q. Notice that the problem does NOT say that p is a proper subset of r!