"The following are equivalent" What does it mean for several statements to be equivalent? And why does it suffice to prove a "cyclic" chain
$$A_1\implies A_2\implies \cdots\implies A_n\implies A_1$$
in order to show that the conditions $A_1, \dots, A_n$ are equivalent?

Two statements are equivalent if
$$A_1\Rightarrow A_2\text{ and } A_2\Rightarrow A_1.$$
More than two statements are equivalents if any two of them are equivalent.
So if you have for instance 
$$A_1\Rightarrow A_2\Rightarrow A_3\Rightarrow A_1$$
then
$$A_2\Rightarrow A_1$$
so $A_1\iff A_2$. In the same way you could show that $A_2\iff A_3$ and $A_1\iff A_3$.
It is the basic idea, and it is working for any chain of length $n$ (you can prove it by induction).
