Test Review 1 - lim sups and lim infs

In summary, the conversation is about a student asking for help with a question regarding a test in Advanced Calculus. The student provides some initial thoughts and asks for suggestions. Another person named Colleen offers a suggestion to compare the expressions using the fact that lim inf sn = -lim sup -sn. The student is unsure if this would be helpful.
  • #1
cmurphy
30
0
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
 
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  • #2
Would it be at all helpful to look at the fact that lim inf sn = -lim sup -sn, or is that just complicating matters?

Colleen
 

1. What is a lim sup and lim inf?

A lim sup (limit superior) is the largest limit that a sequence can approach, while a lim inf (limit inferior) is the smallest limit that a sequence can approach. They are used to determine the limit behavior of a sequence.

2. How do you calculate lim sup and lim inf?

To calculate the lim sup and lim inf of a sequence, you take the set of all possible limits of the sequence and find the smallest and largest values, respectively. If these two values are equal, then the sequence has a limit. If they are not equal, then the sequence does not have a limit.

3. What is the difference between lim sup and lim inf?

The main difference between lim sup and lim inf is that lim sup is the largest limit that a sequence can approach, while lim inf is the smallest limit that a sequence can approach. Additionally, lim sup is the limit as n approaches infinity from above, while lim inf is the limit as n approaches infinity from below.

4. When is a sequence said to converge?

A sequence is said to converge if both its lim sup and lim inf exist and are equal. In this case, the limit of the sequence is equal to the lim sup and lim inf value.

5. How are lim sup and lim inf used in real life?

Lim sup and lim inf are used in various fields of science and mathematics, such as calculus, statistics, and physics. In real life, they are used to analyze and predict the behavior of complex systems, such as the stock market or weather patterns.

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