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xeno_gear

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## Homework Statement

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

## Homework Equations

There's a suggestion in the text:

If [tex]\sum_na_n^2 = \infty[/tex], then there are disjoint sets [tex]E_k \, (k=1,2,3,\dots)[/tex] so that

[tex]\[\sum_{n\in E_k}a_n^2 > 1.\][/tex]

Define [tex]b_n[/tex] so that [tex]b_n = c_ka_n[/tex] for [tex]n \in E_k[/tex]. For suitably chosen [tex]c_k[/tex], [tex]\sum_na_nb_n = \infty[/tex] although [tex]\sum_nb_n^2 < \infty.[/tex]

## The Attempt at a Solution

Using the hint, we've got

[tex]

\[b_n^2 = c_k^2a_n^2; \qquad\qquad a_nb_n = c_ka_n^2\]

\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*}

[/tex]

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