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:

(1) There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\sum\limits_{n=1}^{\infty} \dfrac{a_n}{c_n}$ and $\sum\limits_{n=1}^{\infty} \dfrac{c_n}{b_n}$ both converge;

(2) $\sum\limits_{n=1}^{\infty} \sqrt{\dfrac{a_n}{b_n}}$ converges.