Completeness of the real numbers

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Related lectures (31)

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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.