Complete metric spaceIn mathematical analysis, a metric spaceM is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).
Limit of a sequenceIn mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the lim{\displaystyle \lim } symbol (e.g., limn→∞an{\displaystyle \lim _{n\to \infty }a_{n}}). If such a limit exists and is finite, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
SequenceIn mathematics, a sequence is a collection of objects possibly with repetition, that come in a specified order. Like a set, it contains members (also called elements, or terms). Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.
Convergent seriesIn mathematics, a series is the sum of the terms of an infinite sequence of numbers.
Augustin-Louis CauchyBaronAugustin-Louis Cauchy (21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real analysis), pioneered the field of complex analysis, and the study of permutation groups in abstract algebra. Cauchy also contributed to a number of topics in mathematical physics, notably continuum mechanics.Uniform spaceTopological space with a notion of uniform propertiesIn the mathematical field of topology, a uniform space is a set with additional structure that is used to define uniform properties, such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis.
Net (mathematics)Generalization of a sequence of pointsIn mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize the concept of a sequence in a metric space.
Iterative methodIn computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the i-th approximation (called an "iterate") is derived from the previous ones.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
AlgorithmIn mathematics and computer science, an algorithm () is any well-defined set of instructions that when followed terminates after a finite number of steps that comprise a solution to a given computational problem. Advanced algorithms may utilize loops and involve many conditionals that decide the next step based on the inputs provided, resulting in long sequences of steps before halting, but all algorithms terminate by definition.
Element of a setIn mathematics, an element (or member) of a set is any one of the distinctobjects that belong to that set. For example, given a set called A containing the first four positive integers(A={1,2,3,4}{\displaystyle A=\{1,2,3,4\}}), one could say that "3 is an element of A", expressed notationally as 3∈A{\displaystyle 3\in A}. SetsWriting A={1,2,3,4}{\displaystyle A=\{1,2,3,4\}} means that the elements of the set A are the numbers 1, 2, 3 and 4.*As an Amazon Associate I earn from qualifying purchases.