What Are the Subsequential Limit Points of the Sequence X_n = cos(n)?

[CyberShot]

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!

 

[Mark44]

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?

 

[CyberShot]

The definition of what?

The definition of subsequential limit point.

 

[Mark44]

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.

 

[Bacle2]

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.