IMC 2018 Problem 1

Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent: [list=1] []There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}$ and $\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}$ both converge;[/] [] $\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}$ converges.[/] [/list]

Proposed by Tomáš Bárta, Charles University, Prague