H be a subgroup of G. Show C(C(C(H)))) = C(H)

hint: show if A is a subgroup of B then C(A) is a subgroup of C(B).

I started off trying to prove the hint.

If A is a subgroup of B then I let there intersection be K and then C(K) has only elements of C(A) but they are also contained in C(B) therefor C(A) is a subgropu of C(B).

Not sure if this is correct and if it is not sure how to apply this to the main question.

Any help greatly apreciated