Series question (non-baby rudin, ch. 4, #7)

[xeno_gear]

Homework Statement

Suppose $$\{a_n\}$$ is a sequence of positive numbers such that $$\sum_na_nb_n < \infty$$ whenever $$b_n \ge 0$$ and $$\sum_nb_n^2 < \infty$$. Prove that $$\sum_na_n^2 < \infty$$.

Homework Equations

There's a suggestion in the text:
If $$\sum_na_n^2 = \infty$$, then there are disjoint sets $$E_k \, (k=1,2,3,\dots)$$ so that
$$\[\sum_{n\in E_k}a_n^2 > 1.\]$$
Define $$b_n$$ so that $$b_n = c_ka_n$$ for $$n \in E_k$$. For suitably chosen $$c_k$$, $$\sum_na_nb_n = \infty$$ although $$\sum_nb_n^2 < \infty.$$

The Attempt at a Solution

Using the hint, we've got
$$\[b_n^2 = c_k^2a_n^2; \qquad\qquad a_nb_n = c_ka_n^2\] \begin{align*} S1 = \sum_n b_n^2 &= \sum_{n\in E_1} c_1^2 a_n^2 + \sum_{n\in E_2} c_2^2 a_n^2 + \dots \\ &= \sum_k \sum_{n_k\in E_k} c_k^2 a_{n_k}^2 < \infty\\ S2 = \sum_n a_n b_n &= \sum_k \sum_{n_k\in E_k} c_k a_{n_k}^2 = \infty. \end{align*}$$

From this, it seems clear that we want to choose the $$c_k$$ to be "sufficiently dampening" so that $$S1$$ really is finite but $$S2$$ is not. I'm not entirely sure how to go about choosing the $$c_k$$ for arbitrary $$a_n$$ and $$b_n$$. Thanks very much, any hints would be greatly appreciated! Sorry about the formatting, I'm new here..

 

[statdad]

Perhaps a well-known inequality dealing with sums would help.

 

[xeno_gear]

Thanks for the hint! I'm assuming you're referring to the CBS inequality, but I'm not quite sure how to apply it in a useful way.

I've got
$$\sum_n a_nb_n \le \sqrt{\sum_n a_n^2}\sqrt{\sum_n b_n^2}$$
and the RHS is infinite since the sum of the squares of the a_n is infinite.