Lecture

Real Numbers Sequences

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

In course

MATH-101(d): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable. ...

Description

This lecture covers the definitions and examples of real number sequences, including the notation and properties of sequences. It also discusses the convergence of sequences, the concepts of boundedness, and the limits of sequences. The lecture emphasizes the importance of understanding the convergence and boundedness of sequences in real analysis.

Official source

Ontological neighbourhood

Mathematics

Topology: General topology

Related concepts (32)

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

Construction of the real numbers

In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition. The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them.

Definable real number

Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, , can be defined as the unique positive solution to the equation , and it can be constructed with a compass and straightedge. Different choices of a formal language or its interpretation give rise to different notions of definability.

Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system by adding two infinity elements: and where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted or or It is the Dedekind–MacNeille completion of the real numbers.

Real analysis

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.

Related lectures (64)

Real Numbers Sequences

Explores real number sequences, convergence, limits, boundedness, and monotonicity.

Limit of a Sequence MOOC: Analysis I (part 3) : Real number sequences I and II MOOC: Analysis I

Explains the concept of the limit of a sequence and its importance in convergence.

Limits of Sequences

Explores the concept of limits of sequences and their convergence towards infinity or negative infinity.

Real Analysis: Sequences and Limits

Covers real sequences, induction, limits, and convergence in mathematical analysis.

Sequence Convergence: Definitions and Properties