Question: The product of monotone sequences is monotone, T or F?
Uncompleted Solution: There are four cases from considering each of two monotone sequences, increasing or decreasing.
CASE I: Suppose we have two monotonically decreasing sequences, say ${\{a_n}\}$ and ${\{b_n}\}$. Then, $a_{n+1}\leq a_n$ and $b_{n+1}\leq b_n$; if $b_n\geq 0$ and $b_{n+1}\geq 0$ then $a_{n+1}b_{n+1}\leq a_{n}b_{n+1}\leq a_{n}b_{n}$, but the l-h-s inequality, i.e., $a_{n}b_{n+1}\leq a_{n}b_{n}$, implies that must $a_{n}\geq 0$ since $b_{n+1}\leq b_n$ already has been supposed, but $a_{n}\geq 0$ has not been supposed. So does it mean that two monotonically decreasing sequences with requisites of $a_{n}\leq 0$ since $b_{n+1}\leq b_n$; is counterexample for "the product of monotone sequences is monotone"?
Under which circumstances the product of monotone sequences is monotone, even if it may not true for all cases? And, is there any short (general) proof without need to evaluate each single of sub-cases of the 4-cases?
Thank you.