Concept

Completeness of the real numbers

Related lectures (31)

Explores the fundamental concepts of real numbers, including sets, operations, and properties like supremum and infimum.

Real Analysis Basics

Covers the basics of real analysis, including limits, sequences, and continuous functions.

Introduction to Real Numbers

Introduces the properties and structure of real numbers, emphasizing completeness and the Archimedean property.

Properties of Real Numbers

Explains the properties of subsets of real numbers, including Supremum, Infimum, intervals, open sets, and closed sets.

More Results on Inf/Sup, Density of Q in R

Covers inf/sup, integral part of real numbers, and density of rational numbers.

Absolute Value and Intervals

Introduces absolute value, intervals, triangle inequality, and complete numbers in the complex plane.

Real Numbers: Sets and Operations

Covers the basics of real numbers and set theory, including subsets, intersections, unions, and set operations.

Introduction to Real Numbers

Introduces the axiomatic structure of real numbers and their properties, including completeness and the Archimedean property.

Dedekind Cuts: Rational Numbers

Explores Dedekind cuts in rational numbers, essential for constructing real numbers.

Mathematics: Analysis and Algebra Overview

Provides an overview of analysis and algebra courses, focusing on real numbers, limits, functions, and exams.

Real Numbers: Sets and Operations

Covers the fundamental concepts related to real numbers, including sets, notations, and operations.

Properties of Real Numbers: Bounds, Density, Absolute Value

Covers the properties of real numbers, including bounds, density, and absolute value.

Functions defined by entire series

Covers the concept of functions defined by entire series and their association with functions f(x).

Real Numbers: Order and Completeness

Covers the properties of real numbers, focusing on the total order and completeness, including the Archimedean property and the concepts of supremum and infimum.

Selected Topics in Mathematics

Covers selected topics in mathematics, including Taylor approximations and algebraic structures of Z and K[X].

Real Number Sequences: Cauchy Sequences

Discusses Cauchy sequences, sub-sequences, and numerical series with parameters.

Prelude: Linking Mathematics and Course Content

Introduces the course content by linking familiar mathematics with upcoming topics.

Infimum

Explains the concept of infimum in real numbers and its properties.

Inequality in Real Numbers

Explores the equivalence of inequalities to determine the least upper bound of a set.

Important Property of Archimedean Bodies

Covers the Archimedean property and the proposition for all x in a set, there exists a positive real number n such that n*x is greater than y.