Here is a proof from Atiyah-Macdonald:
For i) $\implies$ ii) could one not write "If $(x_n)$ is such that $x_m = x_{m+1} = \dots$ then obviously $x_m$ is a maximal element"?
I am asking because the book has been getting terser with proofs and has reached super-terse by now but "i) $\implies$ ii)" seems much longer than necessary. So I must be missing something.
Note: for the equivalence of these two statements we need choice (which is not mentioned in the book).