"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?

Suppose we have shown that $A_1 \implies A_2 \implies \cdots \implies A_n \implies A_1$.
Let $t(\varphi) \in \{0,1\}$ denote the truth value of $\varphi$.
Then the infinite sequence given by
$$
t(A_1), t(A_2), t(A_3), \ldots, t(A_n), t(A_1), t(A_2), \ldots t(A_n), t(A_1), \ldots
$$
is periodic and non-decreasing, so it is constant.
That is, either all $A_i$ are true, or all $A_i$ are false.
