Proof: Sequence (-1)^n Diverges | Real Analysis

Support the production of this course by joining Wrath of Math to access all my Real Analysis videos plus lecture notes at the premium tier! https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 🛍 Check out 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/4lNpyNN Business Inquiries: [email protected] We prove the sequence (-1)^n diverges. This is an example of a sequence that diverges, but not to positive or negative infinity. Thus, we will prove it diverges using the plain old definition of divergence - that is, that it does not converge to any real number. To prove (-1)^n doesn't converge to any real number, we will assume for contradiction that it does converge to a real number, and show this forces a contradiction. In a nutshell, by assuming (-1)^n converges to a limit L, this forces the terms of the sequence to get arbitrarily close to L. Since the sequence alternates between -1 and 1, this means -1 should be arbitrarily close to L and so should 1. This is obviously not possible. For example, 1 and -1 should both be within 1/2 of L, but this means L is both positive and negative. We'll see the details of this contradiction in the full lesson. Intro to Sequences: https://www.youtube.com/watch?v=YEiZWonJrOg&list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli&index=14 Definition of the Limit of a Sequence: https://www.youtube.com/watch?v=cTnlHZD5ss4&list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli&index=15 Sequences that Diverge to Infinity (Definition): https://www.youtube.com/watch?v=NjpL3IIsEV4&list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli&index=17 Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu