Understanding lim sup and lim inf: Finding limit points and subsequential limits

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In summary, the conversation discussed the concept of subsequential limits and their relationship to the limsup and liminf of a sequence. It was mentioned that S, the set of all subsequential limits, is also the set of all limit points of the sequence. However, the question of how to determine this set was raised, with an example of s_n=\frac{1}{n} being given. Ultimately, the individual was able to understand the concept.
  • #1
funcalys
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Hi everyone,
I'm currently having problems with these concept, my textbook states that:
Let S be the set of all subsequential limits of s_n, then
sup S= limsup s_n
inf S= liminf s_n
Knowing that S is also the set of all limits point of s_n, however I'm wondering how I could determine this set.
Ex: For s_n=[itex]\frac{1}{n}[/itex], I can easily check that 0 is its limit point but I don't know how to find it.:frown:
 
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  • #2
You say "easily check", and "I don't know how to find it". It's not clear to me where you are stuck. Are you wondering how to show S={0}? If so, how about showing each is a subset of the other?

Please let us know if your question lies somewhere else.
 
  • #3
Never mind, I grasped the idea at last :biggrin: , but thanks aw.
 
Last edited:

What is lim sup and lim inf?

Lim sup and lim inf are mathematical concepts used to describe the behavior of a sequence of numbers. Lim sup (limit superior) is the largest limit that the sequence can approach as the index approaches infinity. Lim inf (limit inferior) is the smallest limit that the sequence can approach.

How do you calculate lim sup and lim inf?

To calculate lim sup, you need to find the largest limit that the sequence can approach as the index approaches infinity. This is often done by finding the limit of the upper bound of the sequence. To calculate lim inf, you need to find the smallest limit that the sequence can approach. This is often done by finding the limit of the lower bound of the sequence.

What is the significance of lim sup and lim inf?

Lim sup and lim inf are important concepts in analysis and calculus. They help us understand the behavior of a sequence and can be used to prove convergence or divergence of a series. They are also useful in finding the limit of a sequence when the limit does not exist.

How do lim sup and lim inf relate to each other?

Lim sup and lim inf are related to each other through the following inequality: lim inf <= lim sup. This means that the limit inferior is always less than or equal to the limit superior. This relationship helps us understand the behavior of a sequence and can be used to prove convergence or divergence.

What are some common applications of lim sup and lim inf?

Lim sup and lim inf are used in various fields such as physics, engineering, and economics. They are used to analyze the behavior of systems and predict future outcomes. They are also important in optimization problems and in understanding the stability of systems.

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