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Let $\sum_{k=1}^{\infty}a_k$ be a series of real numbers. Suppose that $\sum_{k=1}^{\infty}a_kb_k$ converges for every bounded sequence {$b_k$} ? Prove that $\sum_{k=1}^{\infty}a_k$ converges absolutely.

Here is my thought process:

Since {$b_k$} is bounded, every |$b_k$| $\leq$ M for all k.

So $\sum_{k=1}^{\infty}|a_kb_k|$ = $\sum_{k=1}^{\infty}|a_k||b_k|$ $\leq$ $\sum_{k=1}^{\infty}|a_k|M$ = $M$$\sum_{k=1}^{\infty}|a_k$| and since $\sum_{k=1}^{\infty}|a_kb_k|$ converges, $\sum_{k=1}^{\infty}|a_k|$ also converges.

However, I am having a hard time figuring out how to prove absolute convergence. I realize that they are similar questions on this website but they're asking to prove that $\sum_{k=1}^{\infty}a_kb_k$ converges, rather than $\sum_{k=1}^{\infty}a_k$ converging absolutely. Would really love any help, I'm quite stuck!

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  • $\begingroup$ Why'd you drop the absolute value bars...? $\endgroup$ – Simply Beautiful Art Mar 5 '17 at 22:35
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    $\begingroup$ Have you tried looking at a specific sequence $(b_k)_k$, depending on $(a_k)_k$, such that (1) $(b_k)_k$ is bounded, and (2) $a_kb_k = \lvert a_k\rvert$ for every $k$? $\endgroup$ – Clement C. Mar 5 '17 at 22:36
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Hint: define $b_k=\frac{|a_k|}{a_k}$ if $a_k\neq 0$, and $b_k=0$ if $a_k=0$.

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  • $\begingroup$ I'm not sure I understand how to proceed with that... $\endgroup$ – JxxYsde3 Mar 5 '17 at 23:18
  • $\begingroup$ The sequence $\{b_k\}$ in my answer is bounded, and $a_kb_k=|a_k|$. $\endgroup$ – carmichael561 Mar 5 '17 at 23:26

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