TITLE: Let $\sum _{n=1}^{\infty} a_{n} $and $\sum _{n=1}^{\infty} b_{n} $ converge absolutely. Prove that ...
QUESTION [5 upvotes]: 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.

REPLY [12 votes]: Hint: $\sqrt{|a_nb_n|}\leq (|a_n|+|b_n|)/2$.

REPLY [5 votes]: Are you familiar with the Cauchy-Schwarz inequality?
$$
\sum x_i y_i \leq \left(\sum x_i^2 \right)^{1/2} \left(\sum y_i^2\right)^{1/2}.
$$
In this case take $x_i = \sqrt{|a_i|}, y_i = \sqrt{|b_i|}$ and we find
$$
\sum \sqrt{|a_n b_n|} \leq \left(\sum |a_i| \right)^{1/2} \left(\sum|b_i|\right)^{1/2} < \infty.
$$

REPLY [5 votes]: 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.

Since $\sum |a_n|$ and $\sum |b_n|$ converge, $\;\sum (|a_n| + |b_n|)\;$ converges.
Then, essentially, you need only to prove that $$\sqrt{|a_n b_n|} \leq \frac{1}{2}(|a_n| + |b_n|).$$

REPLY [3 votes]: Hint:
1) Note that $\sum |a_n|$ and $\sum |b_n|$ converge.
2) Prove that $\sum (|a_n| + |b_n|)$ converges.
3) Prove that $\sqrt{|a_n b_n|} \leq \frac{1}{2}(|a_n| + |b_n|)$, thus $\sum\sqrt{|a_n b_n|}$ converges.