prove that if A is a subset of B, B is a subset of C, and C is a subset of A, then A=B and B=C To prove A=B, I must prove that A is a subset of B and B is a subset of A. A is a subset of B is already given. So all that is left is to prove B is a subset of A. 
Is it suffice to say that since A is a subset of B, B is a subset of C, and C is a subset of A, by transitive property B is a subset of A. 

In fact, the problem deals wit two separate issues!
Check transitivity:
$$A\subseteq B\subseteq C\implies A\subseteq C$$
Check antisymmetry:
$$A\subseteq C\subseteq A\implies A=C$$
(These are rather different!)
