Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $\eta > 0$. Show that there exists $A \geqslant 0$ such that for all $x > A$, $$\Lambda \cap [x, x + \eta] \neq \varnothing$$
Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $\eta > 0$. Show that there exists $A \geqslant 0$ such that for all $x > A$,
$$\Lambda \cap [x, x + \eta] \neq \varnothing$$