If $a$ is a positive number, show that $\inf \{a/n: n \in \mathbb{Z}^{+} \} = 0$.
So let $A = \{a/n: n \in \mathbb{Z}^{+} \}$. Then $A$ is bounded below by $0$. Hence $\alpha = \inf(A)$ exists. So $0 \leq \alpha$. Now $\alpha \leq a/2n$ which implies that $2 \alpha$ is a lower bound for $A$. Thus $2 \alpha \leq \alpha$. This can only be true if $\alpha \leq 0$. By trichotomy, $\alpha = 0$.
Does this seem correct?