Proving series convergence, conditionally convergent and absolute convergent series 
Give an example of two convergent series $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ so that the series $\sum_{n=1}^\infty a_n b_n$ diverges. Prove that it's not possible to give such an example if we also demand that one of the series $\sum_{n=1}^\infty a_n$ or $\sum_{n=1}^\infty b_n$ shall converge absolutely. 

I was able to find the examples for the first part. For the secound part, it's simple if we assume one of the series converges absolutely and the other one doesen't, because then we know $$\sum_{n=1}^\infty a_n b_n \leq \sum_{n=1}^\infty a_n  $$ if $\sum_{n=1}^\infty a_n$ converges absolutely and $\sum_{n=1}^\infty b_n$ converges conditionally. However i don't know how to prove it if thats not the case.

Your inequality is wrong as you have seen in the comments. But we can do something similar. I'll write $\sum$ for $\sum_{n=1}^\infty$. Without out loss of generality assume $\sum a_n$ converges absolutely. Since $\sum b_n$ converges we have $b_n\to 0$ as $n\to\infty$ and that implies: $b_n$ is bounded. That means there is some $M>0$ such that $|b_n|<M$ for all $n$. We put everything together and get:
\begin{align} 
\sum |a_nb_n| \leq \sum M|a_n|
\end{align} 
So $\sum |a_nb_n|$ convergent by comparison test. Absolute convergence implies convergence and therefore $\sum a_nb_n$ is convergent. 
