Question 19
Let $\varepsilon\gt0$ be given. Show that there exist integers $a, b$ and $c$, not all $0$, such that the inequality $|a \sqrt{2}+b \sqrt{3}+c \sqrt{5}|\lt\varepsilon$ holds.
Prove further that in every triple $(a, b, c) \neq(0,0,0)$ of integers for which this inequality holds, at least one of the absolute values $|a|,|b|,|c|$ is greater than $\varepsilon^{-1 / 3} / \sqrt{30}$.
Prove further that in every triple $(a, b, c) \neq(0,0,0)$ of integers for which this inequality holds, at least one of the absolute values $|a|,|b|,|c|$ is greater than $\varepsilon^{-1 / 3} / \sqrt{30}$.