Proof: Sequence n^2 Diverges to Infinity | 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 a_n = n^2 diverges to positive infinity. It is very important we understand the definitions we'll be using a lot as we study real analysis, and nothing helps us understand definitions quite like proving an object satisfies a certain definition! Today we prove the sequence of squares of natural numbers satisfies the definition of diverging to positive infinity. To prove a sequence diverges to positive infinity, we just have to prove that for any positive real number, say M, our sequence will eventually pass M and stay above M. In other words, we need to show that for every M greater than 0 there exists a real number N so that if n is greater than N then a_n is greater than M. We often figure out how to construct such a proof by beginning with scratchwork: we assume a_n is greater than M and solve for n to determine a restriction that will force the inequality we want - then we use that in our proof. We use this common method in today's divergent sequence proof. Intro to Sequences: https://www.youtube.com/watch?v=YEiZWonJrOg Definition of the Limit of a Sequence: https://www.youtube.com/watch?v=cTnlHZD5ss4 Sequences that Diverge to Infinity: https://www.youtube.com/watch?v=NjpL3IIsEV4 Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu