1. Aug 27, 2012 #1

   CyberShot

   

   1. The problem statement, all variables and given/known data
   
   Determine all subsequential limit points of the sequence X_n = cos(n)
   
   
   2. Relevant equations
   
   Unsure of any.
   
   3. The attempt at a solution
   
   Tried determining subsequences of cos(n) but, having trouble finding any.
   
   
   Can anyone tell me the definition and how to proceed?
   
   Thanks!

    

   CyberShot, Aug 27, 2012
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3. Aug 27, 2012 #2

   Mark44

   

   Insights Author

   Staff: Mentor

   The definition of what?

    

   Mark44, Aug 27, 2012
4. Aug 27, 2012 #3

   CyberShot

   

   The definition of subsequential limit point.

    

   CyberShot, Aug 27, 2012
5. Aug 27, 2012 #4

   Mark44

   

   Insights Author

   Staff: Mentor

   A limit point of a subsequence.
   
   So that it doesn't appear that I'm being flip, here's an example to illuminate this concept. Consider a_n = cos(n * ##\pi/2##), n ≥ 1.
   
   The first few terms of the sequence: {0, -1, 0, 1, 0, -1, 0, 1, ...}
   
   0 is a limit point of the subsequence {0, 0, 0, ... }.
   Likewise, -1 and 1 are limit points of the subsequences {-1, -1, -1, ...} and {1, 1, 1, ...}, respectively.

    

   Mark44, Aug 27, 2012
6. Aug 28, 2012 #5

   Bacle2

   

   Science Advisor

   To give yet another perspective, consider your sequence as just a collection of
   
   points. A collection of points may have more than one limit point ( or, of course,
   
   no limit points or exactly one limit point). Informally, a subsequence is a subset of the
   
   original sequence in which the order of the terms is preserved.

    

   Bacle2, Aug 28, 2012

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