Need help analysing a particular set

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SUMMARY

The discussion centers on the analysis of the set S = (-1)^n * (3 + 5/n). The maximum value of this set is established as 3 + 5/2, while the minimum is -8. It is concluded that the set is not closed because, despite having a maximum and minimum, the limit point 3 is not contained within the set S. This indicates that the set does not meet the criteria for being closed.

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Homework Statement



For the set S = (-1)^n * (3 + 5/n)
I have determined that the maximum is 3 + 5/2 and the minimum is -8. However I am not sure if it is closed; given that it has a maximum and minimum, does this mean by definition that the set will be closed?

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The Attempt at a Solution

 
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The number 3 is a limit point of the set, which means that for any ##\epsilon>0## we have a value ##x\in S## such that ##|x-3|<\epsilon##. However, 3 itself is not contained by the set ##S##. Search with google for "closed set limit points" and you will find the answer to your question.
 

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