In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will be after some instances have passed. The use of the term "eventually" can be often rephrased as "for sufficiently large numbers", and can be also extended to the class of properties that apply to elements of any ordered set.
Notation
The general form where the phrase eventually is found appears as follows: which is actually a shorthand for: or somewhat more formally: This does not necessarily mean that any particular value for is known, but only that such an exists. The phrase "sufficiently large" should not be confused with the phrases "arbitrarily large" or "infinitely large". For more, see Arbitrarily large#Arbitrarily large vs. sufficiently large vs. infinitely large.
For an infinite sequence, one is often more interested in the long-term behaviors of the sequence than the behaviors it exhibits early on. In which case, one way to formally capture this concept is to say that the sequence possesses a certain property eventually, or equivalently, that the property is satisfied by one of its subsequences. For example, the definition of a sequence of real numbers converging to some limit' is: When the term "eventually" is used as a shorthand for "there exists a positive number such that for all ", the convergence definition can be restated more simply as: Here, notice that the set of integers that do not satisfy this property is a finite set; that is, the set is empty or has a maximum element. As a result, the use of "eventually" in this case is synonymous with the expression "for all but a finite number of terms" – a special case of the expression "for almost all terms". At the basic level, a sequence can be thought of as a function with natural numbers as its domain, and the notion of "eventually" applies to functions on more general sets as well—in particular to those that have an ordering with no greatest element. More specifically, if is such a set and there is an element in such that the function is defined for all elements greater than, then is said to have some property eventually if there is an element such that whenever ', has the said property. This notion is used, for example, in the study of Hardy fields, which are fields made up of real functions, each of which have certain properties eventually.
Examples
"All primes above 2 are odd" can be written as "Eventually, all primes are odd.”
Eventually, all primes are congruent to ±1 mod 6.
The square of a prime is eventually congruent to 1 mod 24.
When a sequence or a function has a property eventually, it can have useful implications in the context of proving something with relation to that sequence. For example, in the context of 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. The term "eventually" can be also incorporated into many mathematical definitions to make them more concise. These include the definitions of some types of limits, and the Big O notation for describing asymptotic behavior.
