In a supplement to Rudin's "Principle of mathematical analysis" there was an exercise asking to prove the following statement:
" Let $\{a_n\}$ and $\{bn\}$ be real valued sequences such that $\,a_n \le b_n, \forall n \in \Bbb N\,$; if $\,\sum a_n \,$ does neither converge or diverge to $\, -\infty \,$, then also $\,\sum b_n \,$ does not converge. "
So far i have just been able to show that, if the sequence of partial sums of $a_n$ is not bounded below then the sequence of partial sums of $b_n$ has a positively divergent subsequence, hence the associated series can not converge. However I find myself unable to complete the proof when the sequence of partial sums of $a_n$ is bounded below.
Any help is highly appreciated.