Introduction to Real Analysis, Lecture 2: Completeness Axiom, Counterexamples, Ordered Fields

Real Analysis Course Lecture 2: the Completeness Axiom and Ordered Fields. Real Analysis course textbook ("Real Analysis, a First Course"): https://amzn.to/3421w9I. Amazon Prime Student 6-Month Trial: https://amzn.to/3iUKwdP. (0:00) Lecture summary. (1:21) Review the structure of the argument that sqrt(2) is irrational. (6:29) How do we know sqrt(2) exists? It will be based on an argument relying on the Completeness Axiom. Statement of Completeness Axiom and description of supremums (sups) (least upper bounds). (12:02) Why doesn't the set of rational numbers Q satisfy the Completeness Axiom? (12:44) Every nonempty set of real numbers that is bounded below has an infimum (inf). It can be proved from the ordinary Completeness Axiom. (15:06) Everywhere continuous nowhere differentiable pathological function example. (19:48) Everywhere differentiable function with discontinuous derivative example. (21:44) Unbounded derivative that exists everywhere. (22:55) Chaotic sequence (logistic map) defined by a recursive equation (use NestList on Mathematica). (28:24) Field definition (as a commutative ring with unity in which every nonzero element is a unit). (33:10) Ordered sets and ordered fields. (36:31) Prove that if x is less than y and z is positive, then xz is less than yz (multiplying both sides of an inequality by a positive number does not change the direction of the inequality). (43:24) The rational numbers are defective for calculus. (Q is not complete...there are sets of rational numbers whose suprema are not rational). Example involving pi. (47:47) Idea of how to show sqrt(2) = 2^(1/2) exists. (50:25) Weird formulas involving pi discovered by Euler (including from the Basel problem). (51:53) The Triangle inequality. "Hands On Start to Mathematica": https://amzn.to/2MycspH Real Analysis Playlist:    • Introduction to Real Analysis Course, Lect...   Check out my blog at: https://infinityisreallybig.com/ Follow me on Twitter:   / billkinneymath   Bill Kinney, Bethel University Department of Mathematics and Computer Science. St. Paul, MN. AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.