If $a_n $ is a sequence such that $\sum |a_n||q_n| < \, \infty$ whenever $\sum |q_n| < \, \infty$ then $a_n$ is a bounded sequence. Is this statement true? I used $a_n$ as $n^4$, and $q_n$ as $ n^{-8}$ for counterexample, is it correct? The solution given was that statement is true.
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3Take $q_n = 0$ for a stronger "counterexample". The premise is that for all summable sequences $(q_n)$ the series $\sum \lvert a_n\rvert\lvert q_n\rvert$ converges. Think Banach-Steinhaus. – Daniel Fischer Oct 24 '16 at 18:26
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Or assume that $(a_n)$ is not bounded, choose an increasing sequence $(N(n))$ such that $|a_{N(n)}|\geqslant n^2$ for every $n$ and define $(q_n)$ as $q_{N(n)}=\frac1{n^2}$ for every $n$ and $q_k=0$ if $k$ is not one of the integers $N(n)$, and note that $\sum a_nq_n$ diverges. – Did Oct 24 '16 at 19:04