Real analysis monotone subsequence

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A sequence in the real numbers (Xn) will always contain a monotone subsequence, either increasing or decreasing. The concept of peak points is crucial, as these points indicate where subsequent elements are smaller. If there are infinitely many peak points, the subsequence can be formed from them directly. In cases where peak points are finite, the challenge lies in identifying how to construct a monotone subsequence from the remaining elements. The boundedness of the sequence plays a significant role, as it guarantees the existence of a convergent subsequence, which can be adjusted to meet monotonicity.
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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...
 
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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.
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
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