Analysis Help: Limit Points of a Sequence

[Armbru35]

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!

 

[pwsnafu]

so far I think 0 is a limit point,

Why do you think that?
Consider |A_n|. 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?

 

[Armbru35]

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?

 

[HallsofIvy]

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 (-1)^n2n/(n-4) 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}.