Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Analysis I: Bounded and Divergent Sequences

In course

MATH-101(g): 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 concepts of bounded and divergent sequences, including definitions, criteria for convergence, and strategies for calculating limits. The instructor explains how to determine if a sequence is bounded or unbounded and provides examples to illustrate the concepts.

Instructors (2)

Annalisa Buffa

Lénaïc Chizat

Official source

Related concepts (18)

Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position.

Sequence space

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication.

Bounded function

In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that for all x in X. A function that is not bounded is said to be unbounded. If f is real-valued and f(x) ≤ A for all x in X, then the function is said to be bounded (from) above by A. If f(x) ≥ B for all x in X, then the function is said to be bounded (from) below by B. A real-valued function is bounded if and only if it is bounded from above and below.

Totally bounded space

In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space). The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.

Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: the consecutive terms become arbitrarily close to each other – their differences tend to zero as the index n grows.

Related lectures (30)

Uniform Integrability and Convergence

Explores uniform integrability, convergence theorems, and the importance of bounded sequences in understanding the convergence of random variables.

Operations on Limits

Explores algebraic operations on limits of sequences and discusses the uniqueness of limits and properties of convergent and divergent sequences.

Real Analysis: Sequences and Limits

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

The Cauchy Theorem for Sequences

Explores the Cauchy theorem for sequences, highlighting the importance of convergence and boundedness.

The Bolzano-Weierstrass Theorem MOOC: Analysis I MOOC: Analysis I (part 3) : Real number sequences I and II