Question

If {an} is convergent and {bn} is divergent, then {anbn} is divergent. if true prove it if false give example

Answer 1

he only approach I can think of is using the fact that {an} and {bn} are both bounded as they are convergent then applying it to ∑anbn and saying it is bounded and increasing/dec and monotonic. However, I don't think we are allowed to prove using the Cauchy product and I'm unsure how to go about this approach (if it is even right)

Since ∑∞n=1bn converges, there exists such N that ∀n≥Nbn≤1.

Convergence of ∑∞n=1anbn is equivalent to convergence of ∑∞n=Nanbn.

For all sufficiently large n we have 0<bn<1 so for all sufficiently large n we have 0<anbn<an so

limn→∞supm≥0|∑j=0j=man+jbn+j|≤limn→∞supm≥0|∑j=0j=man|=0.