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

 

[mighty2000]

,

Thank you for posting your question. This is a great example of the importance of understanding the concepts behind the calculations. In this case, we are dealing with the concepts of lim inf and lim sup, which can be a bit tricky to grasp at first.

First, let's define lim inf and lim sup. Lim inf (limit inferior) is the smallest limit point of a sequence, while lim sup (limit superior) is the largest limit point. In simpler terms, lim inf is the smallest value that a sequence gets arbitrarily close to, while lim sup is the largest value that a sequence gets arbitrarily close to.

Now, let's look at the given inequality: (lim inf sn)(lim sup tn) <= lim inf (sntn). We can rewrite this as (lim inf sn)/(lim inf (sntn)) <= lim sup tn. This is where the concept of lim inf and lim sup come into play. Since sn <= 0, lim inf sn must also be <= 0. Similarly, since (sntn) <= 0, lim inf (sntn) must also be <= 0. This means that the left side of the inequality is either 0 or undefined. However, the right side of the inequality is >= 0, since tn >= 0. This means that the inequality holds true, as any number is greater than or equal to 0.

I hope this helps steer you in the right direction for solving this problem. Remember to always think about the concepts behind the calculations and not just the numbers themselves. Good luck on your test!