MATH 242: Analysis 1 (McGill Final)

Currently Playing: Sequence (1^n) Diverges using Subsequences | 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 We prove the sequence (-1)^n diverges by finding two subsequences of (-1)^n that converge to different limits. We previously proved (-1)^n diverges by using a contradiction argument, assuming it does converge to some real number L and showing an absurdity. However, now that we have proven a sequence converges to L if and only if all of its subsequences converge to L, we have a much quicker way of proving divergence for oscillating sequences! #RealAnalysis Note that this technique does not work for divergent sequences that aren't oscillating. If a sequence diverges to +/-infinity, all of its subsequences do as well, and so it will not be possible to find two subsequences with different limits. Intro to Subsequences: https://youtu.be/KqX0mKcV_3A Proof (-1)^n Diverges: https://youtu.be/nEBF2oIS9Dk Sequence Converges if and only if all its Subsequences Do: https://youtu.be/0oRN_pxq2IM Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: https://crayonangel.bandcamp.com/ Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu