## Cute Series Problem

The following problem comes from Walter’s Rudin classic text Real and Complex Analysis. It’s Exercise 7 of Chapter 4 (Elementary Hilbert Space Theory).

Suppose $(a_{n})_{n=1}^{\infty}$ is a sequence of positive numbers such that $\sum_{n=1}^{\infty}a_{n}b_{n}<\infty$ for any nonnegative sequence $(b_{n})_{n=1}^{\infty}\in\ell^{2}$. Then $\sum_{n=1}^{\infty}a_{n}^{2}<\infty$.

Proof. Suppose that $\sum_{n=1}^{\infty}a_{n}^{2}=\infty$. By induction, we can construct a countable collection of disjoint finite sets $\left\{E_{k}\right\}_{k=1}^{\infty}$ which partition $\mathbb{Z}^{\geq 1}$ and satisfy

$\displaystyle 1<\sum_{n\in E_{k}}a_{n}^{2}<\infty$

We will construct a nonnegative sequence $(b_{n})_{n=1}^{\infty}$ such that $\sum_{n=1}^{\infty}a_{n}b_{n}=\infty$, but $\sum_{n=1}^{\infty}b_{n}^{2}<\infty$. For each $k\in\mathbb{Z}^{\geq 1}$, define

$\displaystyle c_{k}:=\dfrac{1}{k}\left(\sum_{n\in E_{k}}a_{n}^{2}\right)^{-1}$

and for $n\in E_{k}$, define $b_{n}:=c_{k}a_{n}$. Since the partial sums of the series $\sum_{n=1}^{\infty}b_{n}^{2}$ are nonnegative, the order of summation is irrelevant, provided the series converges. Thus,

$\begin{array}{lcl}\displaystyle\sum_{n=1}^{\infty}b_{n}^{2}=\sum_{k=1}^{\infty}\sum_{n\in E_{k}}c_{k}^{2}a_{n}^{2}\left(\sum_{n\in E_{k}}a_{n}^{2}\right)^{-2}&<&\displaystyle\sum_{k=1}^{\infty}\sum_{n\in E_{k}}\dfrac{a_{n}^{2}}{k^{2}}\left(\sum_{n\in E_{k}}a_{n}^{2}\right)^{-1}\\&=&\displaystyle\sum_{k=1}^{\infty}\sum_{n\in E_{k}}\dfrac{1}{k^{2}}\\&=&\displaystyle\sum_{k=1}^{\infty}\dfrac{1}{k^{2}}<\infty\end{array}$,

since $\left(\sum_{n\in E_{k}}a_{n}^{2}\right)^{-1}<1$, and

$\displaystyle\sum_{n=1}^{\infty}a_{n}b_{n}=\sum_{k=1}^{\infty}\sum_{n\in E_{k}}\dfrac{a_{n}^{2}}{k}\left(\sum_{n\in E_{k}}a_{n}^{2}\right)^{-1}=\sum_{k=1}^{\infty}\sum_{n\in E_{k}}\dfrac{1}{k}=\sum_{k=1}^{\infty}\dfrac{1}{k}=\infty$

We arrive at a contradiction, which completes the proof. $\Box$