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Lecture

Limit Superior and Inferior: Convergence and Boundedness

Description

This lecture covers the concepts of limit superior and limit inferior for bounded sequences, showing how these values provide useful information about the behavior of the sequence at infinity. It also explains how to associate two numbers, called limit superior and limit inferior, to any bounded sequence, and how these numbers are equal if and only if the sequence converges.

Official source

Ontological neighbourhood

Mathematics

Topology: General topology

Related concepts (32)

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.

Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to and direct limit in . In formulas, a limit of a function is usually written as (although a few authors use "Lt" instead of "lim") and is read as "the limit of f of x as x approaches c equals L".

Limit of a function

Although the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x.

Computer terminal

A computer terminal is an electronic or electromechanical hardware device that can be used for entering data into, and transcribing data from, a computer or a computing system. The teletype was an example of an early-day hard-copy terminal and predated the use of a computer screen by decades. Early terminals were inexpensive devices but very slow compared to punched cards or paper tape for input, yet as the technology improved and video displays were introduced, terminals pushed these older forms of interaction from the industry.

Display device

A display device is an output device for presentation of information in visual or tactile form (the latter used for example in tactile electronic displays for blind people). When the input information that is supplied has an electrical signal the display is called an electronic display. Common applications for electronic visual displays are television sets or computer monitors. Electronic visual display These are the technologies used to create the various displays in use today.

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