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Let $(a_n)_{n\in \mathbb N}$ be a sequence of real numbers with $a_n\ge0$ for all $n\in\mathbb N$ and suppose $\sum_{n=1}^\infty a_n$ converges. Show that if $(b_n)_{n\in\mathbb N}$ is any bounded sequence of real numbers, then $\sum_{n=1}^\infty a_n b_n$ converges absolutely.

The Monotone Convergence Theorem is an obvious choice here. I think I can see the sequence of partial sums is bounded, but how would I show that the it is monotonically increase or decreasing without anymore information on $(b_n)$? Should I divide it into cases?


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up vote 1 down vote accepted

Since $b_n$ is bounded, there exists some $M$ such that $|b_n| \leq M$ for all $n$. Try comparing the non-negative series $\sum_n a_n |b_n|$ and $\sum_n M a_n$.

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Thank you, didn't think of it that way. – Alti Dec 16 '12 at 4:31

$\sum a_n b_n$ converges absolutely if $\sum |a_n b_n|$ converges. You can see why the latter sum is bounded, and its sequence of partial sums is monotonically increasing since only positive terms are being added.

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