Subsequential limit point

  • Thread starter CyberShot
  • Start date
  • #1
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Homework Statement



Determine all subsequential limit points of the sequence X_n = cos(n)


Homework Equations



Unsure of any.

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!
 

Answers and Replies

  • #2
34,876
6,615

Homework Statement



Determine all subsequential limit points of the sequence X_n = cos(n)


Homework Equations



Unsure of any.

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!

The definition of what?
 
  • #3
133
2
The definition of what?

The definition of subsequential limit point.
 
  • #4
34,876
6,615
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 an = 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.
 
  • #5
Bacle2
Science Advisor
1,089
10
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.
 

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