Lecture

Definitions: Basic Properties

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.

Description

This lecture covers the basic properties of real number sequences, including definitions for increasing, strictly increasing, decreasing, strictly decreasing, bounded, majorized, and minorized sequences. It also explains how to demonstrate if a sequence is bounded or not.

In MOOCs (2)

Analysis I (part 3) : Real number sequences I and II

A sequence of real numbers is a function f:N→R . It is usual to write an:=f(n) for the value of f in n. For example, we could define a sequence f(n):=an:=12n, i.e. a0=1,a1=12,a2=14,a3=18,.... . The central concept is that of the limit of a sequence: this is a real number to which, intuitively, the given sequence gets closer and closer. For example, the sequence an given above admits the number zero as its limit. We'll define the concept of limit in a rigorous way and develop methods for establishing the exi

Analysis I

This course covers : Part 1: Prelude, basic concepts, real numbers. Part 2: Introduction to complex numbers. Part 3: Real number sequences I and II. Part 4: Limit of a function, continuous functions. Part 5: Continuous functions and derivable functions, the derivative function. Part 6: Studies of functions, limited developments. Part 7: Indefinite and definite integrals, integration (selected chapters).

Official source

Quiz

Related concepts (14)

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.

Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded if it is, in a certain sense, of finite measure. Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary is a distinct concept: for example, a circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice versa.

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.

Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus, a function defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.

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.

Related lectures (40)

Distributions & Interpolation Spaces

Covers distributions, interpolation spaces, convergence, and the concept of dual spaces.

Limit of a Sequence

Explores the limit of a sequence and its convergence properties, including boundedness and monotonicity.

Continuous functions on a closed bounded interval

Covers limits, range of continuous functions, and uniform continuity on closed intervals.

Real Sequences: Definitions and Operations

Explores real sequences, limits, and algebraic operations on sequences.

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