Is the product of two monotone sequences monotone?

Solution 1:

A simple counter example to "The product of two monotone sequences is a monotone sequence" is the product of the monotone sequence $\{...,-3,-2,-1,0,1,2,3,...\}$ (which you can picture as a sequence of points in the $y$ axes of the the graph of $f(x) =x$) with itself. The product of these sequences is again a sequence viewed as a series of points in the $y$ axis of $f(x)=x^2$, and is increasing for $n>0$ but decreasing for $n<0$, as can be easily checked. So this shows that even when both sequences are increasing, their product need not be monotone. However, one can easily check that if the sequences are both increasing or both decreasing, and neither change sign, their product is monotone.

Solution 2:

In general the answer is no. Take $a_n = \left ( \frac{5}{4} \right )^n$ and $b_n = \frac{1}{n}$. We then have

\begin{eqnarray*} a_1b_1 & = & \frac{5}{4} \;\; = \;\; 1.25 \\ a_2b_2 & = & \frac{25}{32} \;\; \approx\;\; 0.781 \\ a_3b_3 & = & \frac{125}{192} \;\; \approx \;\; 0.651 \\ a_{10}b_{10} & \approx & 0.93 \\ a_{15}b_{15} & \approx & 1.894. \end{eqnarray*}

We therefore have that neither $a_nb_n \leq a_{n+1}b_{n+1}$ for all $n$, nor $a_n b_n \geq a_{n+1}b_{n+1}$ for all $n$.