Lecture

Real Numbers: Definitions and Bounds

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In course

MATH-101(en): Analysis I (English)

We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable. ...

Description

This lecture covers the definitions and properties of maximum, minimum, supremum, and infimum in the context of real numbers. It explains the notation of number sets, intervals, and bounded intervals, emphasizing the importance of upper and lower bounds. The lecture also discusses the uniqueness of maximum and minimum values, the existence of infimum, and the Archimedean property of real numbers.

Instructor

Thomas Mountford

Teaching & PhD Past EPFL PhD Students Damien Gauthier, Samir Brahim Belhaouari, Driss Baraka, Daniel Rodrigues Valesin, Samuel Schöpfer, Kamran Kalbasi, Jacques Saliba, Hao Xue, Keming Chen Past EPFL PhD Students as codirector Guillaume Charles Woessner Courses Analysis I MATH-101(d) Study the fundamental concepts of analysis, and the differential and integral calculus of real-valued functions of one real variable. Markov chains MATH-332 The course follows the text of Norris

Official source

Related concepts (32)

Infimum and supremum

In mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is the greatest element in that is less than or equal to each element of if such an element exists. In other words, it is the greatest element of that is lower or equal to the lowest element of . Consequently, the term greatest lower bound (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to each element of if such an element exists.

Upper and lower bounds

In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S. Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S. A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound.

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.

Essential infimum and essential supremum

In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, that is, except on a set of measure zero. While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is greater than or equal to the function values everywhere while ignoring what the function does at a set of points of measure zero.

Limit inferior and limit superior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.

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