Let $\sum _{n=1}^{\infty} a_{n} $and $\sum _{n=1}^{\infty} b_{n} $ converge absolutely. Prove that $\sum_{n=1}^{\infty} \sqrt{|a_{n}b_{n}|} $ converges.
I know that the series $\sum_{n=1}^{\infty} a_{n}b_{n}$ converges absolutely, but am having trouble showing what they want. I have tried showing the partial sums are bounded but no luck so far.