I’m stuck on an easy question because… I don’t know. I’m old? Anyway, it’s annoying me, because I usually don’t find these questions too difficult, and I’m pretty sure I’ve seen it before.

If $ frac{b_n}{a_n} overset{ntoinfty}{longrightarrow} alphaneq0, $ then $ displaystylesum a_n $ converges $ iff displaystylesum b_n $ converges

No way this can be false, since at some point $ b_n $ will be tied relatively closely to $ alpha a_n, $ ( i.e. $ b_n $ will be relatively far away from $ 0 $ compared with the distance from $ b_n $ to $ alpha a_n. ) $ I would like to solve it all in one go, i.e. avoiding splitting into cases like $ alpha < 0 $ and then $ alpha > 0, $ because due to “$ b_n $ will be bounded relatively closely to $ alpha a_n$“, this seems to me like it should be straightforward.

I’ve tried a few avenues to go down with regards to a proof: mainly starting from $ varepsilon-n $ definitions of convergence, but before I spend more time on this, *I just want to check if I’m missing something really obvious*.