Is the Absolute Convergence Theorem Sufficient to Prove This?

[alexmahone]

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

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?

 

[tarantella]

Yes it's OK. Just one thing we have in general $|a_n|\cdot |b_n|\leq |a_n|\cdot B$ in general (this inequality is not strict in general).