Скачать или смотреть The Completeness Axiom - Real Analysis | Lecture 3

In this lecture we amend the axioms of an Archimedean ordered field to include the completeness axiom. We see that the addition of this axiom distinguishes the field of rational numbers from the field of real numbers, since only the latter satisfies the completeness axiom. This is exemplified with a Cauchy sequence of rational approximations of the square root of 2, which is proven to not be a rational number. We also define least upper bounds and greatest lower bounds, for which the completeness axiom is shows that bounded sets possess such bounds. Finally, we define the supremum and the infimum of a set, which extends least upper and greatest lower bounds to unbounded sets.

This course is taught by Jason Bramburger.