Let $|\cdot| : \mathbb{R} \longrightarrow \mathbb{R}_{\geq 0}$ be a function such that for every $x, y \in \mathbb{R}$, (i) $|x| = 0$ if and only if $x = 0$; (ii) $|x + y| \leq |x| + |y|$; (iii) $|xy| = |x||y|$. Show that the following are equivalent:\\
(A) The set $\{|n| : n \in \mathbb{Z}\}$ is bounded;\\
(B) $|x + y| \leq \max\{|x|, |y|\}$ for every $x, y \in \mathbb{R}$.