"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?

If you have shown $$A_1\Longrightarrow A_2 \Longrightarrow\ldots\Longrightarrow A_n $$
To show equivalence of all of these statements it suffices to finally show $A_n \Longrightarrow A_1$, because the former implications all hold and so you get equivalence between any two statements. 
For instance, $A_2\iff A_5$ because $A_2\Longrightarrow A_3\Longrightarrow\ldots\Longrightarrow A_5$ means you have $A_2\Longrightarrow A_5$
but by completing the cycle you also have $A_5\Longrightarrow A_6\Longrightarrow\ldots\Longrightarrow A_n\Longrightarrow A_1\Longrightarrow A_2$ which is $A_5\Longrightarrow A_2$.
