Test Review 1 - lim sups and lim infs

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SUMMARY

The discussion focuses on proving the inequality (lim inf sn)(lim sup tn) <= lim inf (sntn) under the conditions that sn <= 0 and tn >= 0 for n in N. The participant, Colleen, establishes that lim inf sn is less than or equal to 0 and lim sup tn is greater than or equal to 0, leading to the conclusion that (lim inf sn)(lim sup tn) is also less than or equal to 0. However, Colleen struggles with comparing the two sides of the inequality since both are non-positive, and seeks guidance on how to proceed with the proof.

PREREQUISITES
  • Understanding of limits, specifically lim sup and lim inf in sequences.
  • Familiarity with the properties of non-positive sequences.
  • Basic knowledge of inequalities in calculus.
  • Experience with advanced calculus concepts and proofs.
NEXT STEPS
  • Study the properties of lim sup and lim inf in detail.
  • Review proofs involving products of sequences, particularly in the context of limits.
  • Explore examples of sequences that illustrate the behavior of lim inf and lim sup.
  • Learn about the implications of sequences approaching zero and their limits.
USEFUL FOR

Students in advanced calculus courses, particularly those preparing for tests on limits and sequences, as well as educators looking for examples of limit proofs involving inequalities.

cmurphy
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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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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
 

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