Generalization of the comparison test (?)

Question

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.

Answer 1

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.