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

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The discussion focuses on understanding the concepts of lim sup and lim inf in relation to subsequential limits and limit points. The user initially struggles to determine the set of subsequential limits for the sequence s_n = 1/n, specifically identifying its limit point. Clarification is sought on how to show that the set of limit points S equals {0}. Ultimately, the user expresses that they have grasped the concept after receiving guidance. This highlights the importance of understanding the relationship between subsequential limits and limit points 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.
 
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