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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

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# Test Review 1 - lim sups and lim infs

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