Sequences satisfying strict inequalities and conditions for limsup = liminf

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In summary, the conversation discusses the search for an example of two sequences that satisfy a set of strict inequalities, as well as the question of when the limit supremum is equal to the limit infimum. The individual is seeking help in finding these sequences and expresses gratitude for any ideas or assistance.
  • #1
PvtBillPilgrim3
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This isn't a copy of my assignment, but I found the exact same question on the web:

http://www.math.auckland.ac.nz/~waldron/730/1.pdf

It's question 1 on the PDF. I've been trying all night to find inequalties that are strict like it asks for in that particular problem. I don't need help with proving the inequalities, just an example of two sequences (xn) and (yn) that satisfy them to make them strict simultaneously.

I would super appreciate any ideas on this.
I just need help where it says: Find sequences for which all of these inequalities are simultaneously strict. I can't find an example.
 
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  • #2
Under what conditions does limsup = liminf?
 

1. What is the definition of liminf and limsup?

The limit inferior (liminf) of a sequence is the smallest limit point that the sequence approaches or a value that is approached infinitely often. The limit superior (limsup) is the largest limit point or value that the sequence approaches.

2. How do you calculate liminf and limsup?

To calculate liminf, you first take the infimum (greatest lower bound) of the set of all subsequential limits of the sequence. To calculate limsup, you take the supremum (least upper bound) of the set of all subsequential limits.

3. What is the difference between liminf and limsup?

The main difference between liminf and limsup is that liminf represents the lower bound of a sequence and limsup represents the upper bound. In other words, liminf is the smallest value that a sequence approaches infinitely often, while limsup is the largest value.

4. Why is liminf and limsup important in mathematics?

Liminf and limsup are important because they provide information about the behavior and convergence of a sequence. They can help determine if a sequence has a limit or if it is divergent. They are also used in the proofs of many theorems in mathematical analysis.

5. Can liminf and limsup be equal?

Yes, liminf and limsup can be equal. This occurs when a sequence has a limit. In this case, the liminf and limsup are both equal to the limit of the sequence. However, if a sequence does not have a limit, then liminf and limsup will be different values.

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