MathDB

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:
[*]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.[/*]
Proposed by Tomáš Bárta, Charles University, Prague

Problem Statement