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#### Alexmahone

##### Active member

- Jan 26, 2012

- 268

Prove: if $\sum a_n$ is absolutely convergent and $\{b_n\}$ is bounded, then $\sum a_nb_n$ is convergent.

$|b_n|\le B$ for some $B\ge 0$.

$|a_n||b_n|<B|a_n|$

Since $\sum|a_n|$ converges, $\sum|a_n||b_n|=\sum|a_nb_n|$ converges.

So, $\sum a_nb_n$ converges. (Absolute convergence theorem)

Is that okay?

**:**__My working__$|b_n|\le B$ for some $B\ge 0$.

$|a_n||b_n|<B|a_n|$

Since $\sum|a_n|$ converges, $\sum|a_n||b_n|=\sum|a_nb_n|$ converges.

So, $\sum a_nb_n$ converges. (Absolute convergence theorem)

Is that okay?

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