Product of 2 Increasing Sequences Not Necessarily Increasing

[Easy_as_Pi]

Homework Statement

Give an example to show that it is not necessarily true that the product of two eventually increasing sequences is eventually increasing.

Homework Equations

a sequence is eventually increasing if for N\in natural numbers, a_{n+1} \geqa_{n} for all n>N.

The Attempt at a Solution

So, I know this is merely proof by counterexample. I find one example to show that the product of two eventually increasing sequences is not necessarily eventually increasing. The only catch is that I have no idea where to start. There are infinitely many eventually increasing sequences I could multiply together. I know the end goal is to show that a_{n+1} - a_{n} is decreasing or eventually decreasing for all n.
So, ideally, I'd end up with something like -x^{2} after a_{n+1} - a_{n}. I don't want a specific example which will solve this problem. Some guidance as to where to begin would be greatly appreciated, though!
Thanks!

 

[Dick]

Try picking an eventually increasing sequence whose values are negative.

 

[Easy_as_Pi]

Sorry that I took an eternity to reply. I was at a study session for linear algebra. Thanks so much, Dick. I think you've helped on every question I've posted here. I really appreciate it. I tripped myself up by only thinking about positive sequences. I took increasing and mistakenly correlated it with positive, too. Immediately after your hint, I thought about -1/n, and then found my solution. This series and sequences course has managed to confuse me more than any previous math class.