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
<br /> \[b_n^2 = c_k^2a_n^2; \qquad\qquad a_nb_n = c_ka_n^2\]<br /> \begin{align*}<br /> S1 = \sum_n b_n^2 &amp;= \sum_{n\in E_1} c_1^2 a_n^2 + \sum_{n\in E_2} c_2^2 a_n^2 + \dots \\<br /> &amp;= \sum_k \sum_{n_k\in E_k} c_k^2 a_{n_k}^2 &lt; \infty\\<br /> S2 = \sum_n a_n b_n &amp;= \sum_k \sum_{n_k\in E_k} c_k a_{n_k}^2 = \infty.<br /> \end{align*}<br />

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
<br /> \sum_n a_nb_n \le \sqrt{\sum_n a_n^2}\sqrt{\sum_n b_n^2}<br />
and the RHS is infinite since the sum of the squares of the a_n is infinite.