Eventually (mathematics)
Encyclopedia
In the mathematical
areas of number theory
and analysis
, an infinite sequence
(an) is said to eventually have a certain property
if the sequence always has that property after a finite number of terms. This can be extended to the class of properties P that apply to elements of any ordered set
(sequences and subsets of R are ordered, for example).
For example, the definition of a sequence of real numbers (an) converging to some limit
a is: for all ε > 0 there exists N > 0 such that, for all n > N, |an - a| < ε. The phrase eventually is used as shorthand for the fact that "there exists N > 0 such that, for all n > N..." So the convergence definition can be restated as: for all ε > 0, eventually |an - a| < ε. In this setting it is also synonymous with the expression "for all but a finite number of terms" - not to be confused with "for almost all
terms" which generally allows for infinitely many exceptions.
A sequence can be thought of as a function with domain the natural number
s. But the notion of "eventually" applies to functions on more general sets, specifically those that have an ordering and no greatest element
. In general if S is such a set and there is an element s in S such that the function f is defined for all elements greater than s, then f is said to have some property eventually if there is an element x0 such that f has the property for all x > x0. This notion is used, for example, in the study of Hardy field
s, which are fields made up of real functions that all have certain properties eventually.
When a sequence or function has a property eventually, it can have useful implications when trying to prove something with relation to that sequence. For example, in studying the asymptotic behavior of certain functions, it can be useful to know if it eventually behaves differently than would or could be observed computationally, since otherwise this could not be noticed. It is also incorporated into many mathematical definitions, like in some types of limits
(an arbitrary bound eventually applies) and Big O notation
for describing asymptotic behavior.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
areas of number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, an infinite sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
(an) is said to eventually have a certain property
Property (philosophy)
In modern philosophy, logic, and mathematics a property is an attribute of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property however differs from individual objects in that...
if the sequence always has that property after a finite number of terms. This can be extended to the class of properties P that apply to elements of any ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
(sequences and subsets of R are ordered, for example).
Motivation and Definition
Often, when looking at infinite sequences, it doesn't matter too much what behaviour the sequence exhibits early on. What matters is what the sequence does in the long term. The idea of having a property "eventually" rigorises this viewpoint.For example, the definition of a sequence of real numbers (an) converging to some limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
a is: for all ε > 0 there exists N > 0 such that, for all n > N, |an - a| < ε. The phrase eventually is used as shorthand for the fact that "there exists N > 0 such that, for all n > N..." So the convergence definition can be restated as: for all ε > 0, eventually |an - a| < ε. In this setting it is also synonymous with the expression "for all but a finite number of terms" - not to be confused with "for almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
terms" which generally allows for infinitely many exceptions.
A sequence can be thought of as a function with domain the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s. But the notion of "eventually" applies to functions on more general sets, specifically those that have an ordering and no greatest element
Greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
. In general if S is such a set and there is an element s in S such that the function f is defined for all elements greater than s, then f is said to have some property eventually if there is an element x0 such that f has the property for all x > x0. This notion is used, for example, in the study of Hardy field
Hardy field
In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that is closed under differentiation. They are named after the English mathematician G. H. Hardy.-Definition:...
s, which are fields made up of real functions that all have certain properties eventually.
When a sequence or function has a property eventually, it can have useful implications when trying to prove something with relation to that sequence. For example, in studying the asymptotic behavior of certain functions, it can be useful to know if it eventually behaves differently than would or could be observed computationally, since otherwise this could not be noticed. It is also incorporated into many mathematical definitions, like in some types of limits
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
(an arbitrary bound eventually applies) and Big O notation
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
for describing asymptotic behavior.