RU-ALLRUSSIAN 1985 Problem 8

$\mathrm{a}_{1}< \mathrm{a}_{2}< \ldots < \mathrm{a}_{\mathrm{n}}< \ldots$ is an unbounded sequence of positive reals. Show that there exists k such that $\mathrm{a}_{1} / \mathrm{a}_{2} + \mathrm{a}_{2} / \mathrm{a}_{3} + \ldots + \mathrm{a}_{\mathrm{h}} / \mathrm{a}_{\mathrm{h} + 1}< \mathrm{h} - 1$ for all $\mathrm{h} > \mathrm{k}$ . Show that we can also find a k such that $\mathrm{a}_{1} / \mathrm{a}_{2} + \mathrm{a}_{2} / \mathrm{a}_{3} + \ldots + \mathrm{a}_{\mathrm{h}} / \mathrm{a}_{\mathrm{h} + 1}< \mathrm{h} - 1985$ for all $\mathrm{h} > \mathrm{k}$ .