Analysis Help: Limit Points

  • Thread starter Armbru35
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  • #1
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Hi, I am asked to find the limit points of the given set A

A={((-1)^n)(2n/(n-4))}

so far I think 0 is a limit point, but I can't figure out if there are any more than just that one.

I believe every point in A is an isolated point, and that this set is open because every point in A contains a neighborhood contained in the set.

Any suggestions would be helpful. Thanks!!
 

Answers and Replies

  • #2
pwsnafu
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so far I think 0 is a limit point,
Why do you think that?
Consider |An|. What happens as n goes to infinity?
I believe every point in A is an isolated point, and that this set is open
If a point is isolated, how can there be an open neighborhood?
 
  • #3
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I guess I thought the limit of infinity/infinity would be 0, but now that I look at it more I feel like the limit would be 2, and because we have the (-1) in front, would the limits be -2 and 2?
 
  • #4
HallsofIvy
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You are using the word "limit" very carelessly. The sequence 2n/(n-4) has limit 2. That means its "set of limit points" is {2}. The sequence [itex](-1)^n2n/(n-4)[/itex] does not converge- it has no limit. It does have two convergent subsequences- taking only even indices, it converges to 2, taking only odd indicees it converges to -2. So its "set of limits points" (not "its limits") is {-2, 2}.
 
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