Monotone Subsequence Theorem (Every Sequence has Monotone Subsequence) | Real Analysis

How nice of a subsequence does any given sequence has? We've seen that not every sequence converges, and some don't even have convergent subsequences. But today we'll prove what is sometimes called the Monotone Subsequence theorem, telling us that every sequence has a monotone subsequence. #RealAnalysis

The key idea of this proof is that of a peak, a term of a sequence that is greater than or equal to all following terms. If a sequence has infinitely many peaks, we can construct a decreasing subsequence of peaks. If a sequence has finitely many peaks, we can construct an increasing subsequence of terms after the last peak of the sequence.

All About Monotone Sequences:    • What are Monotone Sequences? | Real A...  
Proof of Monotone Convergence Theorem:    • Detailed Proof of the Monotone Conver...  
Fun Example of Monotone Convergence Theorem:    • Using the Monotone Convergence Theore...  
If Sequence Diverges to Infinity then so do Subsequences:    • If Sequence Diverges to Infinity then...  
Monotone Sequence Converges if it has Convergent Subsequence:    • Monotone Sequence with Convergent Sub...  

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