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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).
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" rigorizes this viewpoint.
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 numbers. 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 fields, 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.
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