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

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SUMMARY

The discussion focuses on the concepts of limit superior (lim sup) and limit inferior (lim inf) in the context of subsequential limits. It clarifies that the set S, which consists of all subsequential limits of a sequence s_n, is equivalent to the set of all limit points of s_n. The example provided, s_n = 1/n, illustrates that the limit point is 0, and participants emphasize the importance of understanding how to derive this set S effectively.

PREREQUISITES
  • Understanding of sequences and limits in real analysis
  • Familiarity with the definitions of limit points
  • Knowledge of subsequences and their properties
  • Basic mathematical proof techniques
NEXT STEPS
  • Study the formal definitions of lim sup and lim inf in real analysis
  • Explore examples of sequences with multiple limit points
  • Learn about the relationship between subsequences and convergence
  • Investigate the concept of compactness in relation to limit points
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Students of real analysis, mathematicians, and anyone seeking to deepen their understanding of limit points and subsequential limits in sequences.

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=\frac{1}{n}, I can easily check that 0 is its limit point but I don't know how to find it.:frown:
 
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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.
 
Never mind, I grasped the idea at last :biggrin: , but thanks aw.
 
Last edited:

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