# Generalization of the comparison test (?)

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.

Here's a hint based on your work so far: The sequence with terms $$c_{n} := b_{n} - a_{n}$$ is non-negative, so it is either summable or its partial sum are unbounded above.
• If $$(c_{n})$$ is summable, and $$(a_{n})$$ is not,
then $$(b_{n}) = (a_{n} + c_{n})$$ is not;
• If $$(c_{n})$$ is not summable, and $$(a_{n})$$ has partial sums bounded below,
then $$(b_{n}) = (a_{n} + c_{n})$$ has partial sums unbounded above.
• Thank you so much for the help and the elegant answer! I did think about using $b_n-a_n$ but didn’t know how. Feb 12, 2022 at 15:13