Test Review 1 - lim sups and lim infs

cmurphy

Hello,

I am taking Adv. Calc, and we have a test next week. I am going to post a few questions that I have from the review where I got stuck. If you have any help, please steer me in the right direction!

Question 1: Suppose sn <= 0 <= tn for n in N. Prove
(lim inf sn)(lim sup tn) <= lim inf (sntn), provided none of these products is of the form 0 * infinity.

Here is what I have so far:
Since sn <= 0, we must have lim inf sn <= 0.
Also, since tn >= 0, we must have lim sup tn >= 0.
Thus (lim inf sn)(lim sup tn) <= 0.

We also know that (sntn) <= 0.
This means that lim inf (sntn) <= 0.

I am having difficulties at this point, because the two things that I want to compare are both <= 0, so I don't have a way of comparing them.

I'm not sure where to go with this. Any suggestions?
Colleen