Real analysis monotone subsequence

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SUMMARY

The discussion centers on proving that any sequence of real numbers (Xn) contains a monotone subsequence. A sequence is defined as monotone if it is either non-decreasing or non-increasing. The participants highlight the significance of peak points within the sequence, noting that if there are infinitely many peak points, the subsequence can be formed from these points. In cases where the sequence is bounded, it converges, thus guaranteeing the existence of a monotone subsequence.

PREREQUISITES
  • Understanding of real analysis concepts, specifically sequences and subsequences.
  • Familiarity with the definition of monotone sequences.
  • Knowledge of convergent sequences and their properties.
  • Basic comprehension of peak points in sequences.
NEXT STEPS
  • Study the Bolzano-Weierstrass theorem to understand the properties of bounded sequences.
  • Learn about the construction of monotone subsequences from given sequences.
  • Explore the concept of peak points in more depth and their implications in real analysis.
  • Investigate the relationship between bounded sequences and convergence in real analysis.
USEFUL FOR

Students of real analysis, mathematicians focusing on sequence convergence, and educators teaching the properties of monotone subsequences in sequences of real numbers.

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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.
 

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