MATH 242: Analysis 1 (McGill Final)

Currently Playing: Monotone Subsequence Theorem (Every Sequence has Monotone Subsequence) | Real Analysis

Support the production of this course by joining Wrath of Math to access all my real analysis videos plus the lecture notes at the premium tier! https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 🛍 Get the coolest math clothes in the world! https://mathshion.com/ Real Analysis course: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC Get the textbook! https://amzn.to/3CMdgjI 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: https://youtu.be/E_vaJWaeNSU Proof of Monotone Convergence Theorem: https://youtu.be/4NURDmE79VU Fun Example of Monotone Convergence Theorem: https://youtu.be/AuvBS7UKxqU If Sequence Diverges to Infinity then so do Subsequences: https://youtu.be/gL3MRsESCvU Monotone Sequence Converges if it has Convergent Subsequence: https://youtu.be/BZ-LQpz5EBc Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu