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I have the following question, whose inverse question can be done by the well-known Cauchy–Schwarz inequality. But I do not know how to solve this question:

Suppose that $\{a_n\}_{n=1}^\infty$ is a sequence of real numbers such that \begin{equation} \sum_{n=1}^\infty a_n b_n \quad \text{is a convergent series whenever}\quad\sum_{n=1}^\infty b_n^2<\infty. \end{equation} Show that $\sum_{n=1}^\infty a_n^2<\infty$.

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marked as duplicate by Hans Lundmark, JMP, Xam, Claude Leibovici, Lord Shark the Unknown Aug 27 '17 at 5:36

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Another one: Prove that a sequence is square summable. $\endgroup$ – Martin R Aug 19 '17 at 21:05
  • $\begingroup$ Indeed this question has answers in other same posts but i don't think they answer effectively the O.P's question in case the O.P is not familiar with functional analysis..I beleive he looks for an elementary solution if it exists $\endgroup$ – Marios Gretsas Aug 19 '17 at 21:08
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    $\begingroup$ @MariosGretsas: OP did not provide any context (which is generally unfortunate) and did not care to update the question after the close vote, so we cannot guess what he is looking for. – The second Q&A that I linked to is tagged [real-analysis] as well, which makes it a suitable duplicate target in my opinion. Adding another (simpler/better) answer to the other thread is always an option. $\endgroup$ – Martin R Aug 20 '17 at 15:13
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    $\begingroup$ @MartinR that was my opinion..i judged from the question ,reputation of the O.P and froms his posts in his profile conluding that the O.P maybe not familiar with functional analysis techniques to see the answers you proposed..I'm not the type of the user who will rush to close a post down..except the case where the post is the type 'Do my homework now'..But here the o.p posted also a thought of his. $\endgroup$ – Marios Gretsas Aug 20 '17 at 15:45
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Proof by contraposition. Suppose $\sum a_n^2 = \infty$. Using the Cauchy criterion, show that there is a positive number $C$ and a sequence $\{F_k\}$ of disjoint finite subsets of $\Bbb N$ such that $$\sum_{n\in F_k} a_n^2 > C^2\quad (k = 1,2,3,\ldots)$$ Set $s_k = \sum\limits_{n\in F_k} a_n^2$, for $k = 1,2,3,\ldots$. Define $\{b_n\}$ by setting $b_n = \dfrac{a_n}{k\sqrt{s_k}}$ for $n\in F_k$, and $b_n = 0$ if $n$ is not in any of the $F_k$. The sum $$\sum b_n^2 = \sum_k \frac{1}{k^2}\sum_{n\in F_k} \frac{a_n^2}{s_k} = \sum_k \frac{1}{k^2} < \infty$$ and the sum $$\sum a_nb_n = \sum_k \frac{1}{k}\sum_{n\in F_k} \frac{a_n^2}{\sqrt{s_k}} > \sum_k \frac{C}{k} = \infty$$

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