# Product of two eventually increasing sequences which is not eventually increasing

1. Feb 1, 2012

### Easy_as_Pi

1. The problem statement, all variables and given/known data
Give an example to show that it is not necessarily true that the product of two eventually increasing sequences is eventually increasing.

2. Relevant equations
a sequence is eventually increasing if for N$\in$ natural numbers, a$_{n+1}$ $\geq$a$_{n}$ for all n>N.

3. 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!

2. Feb 1, 2012

### Dick

Try picking an eventually increasing sequence whose values are negative.

3. Feb 1, 2012

### 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.