Real analysis monotone subsequence

[Scousergirl]

Homework Statement

Prove:
Let (Xn) be a sequence in R (reals). Then (Xn) has a monotone subsequence.

Homework Equations

Def: Monotone: A sequence is monotone if it increases or decreases.

The Attempt at a Solution

I know it has something to do with peak points...that is there are elements in (Xn) which are peak points (every element afterwards is smaller). There are either an infinite number of peak points (in which case the subsequence consists of the peak points) of finite. I am having a hard time grasping what the subsequence consists of if there are a finite number of peak points...

 

[StatusX]

If the sequence is unbounded, the result is easy. If it is bounded, it has a convergent subsequence. See if you can make this into a monotone subsequence.