Eventually (mathematics)

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 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 (such as sequences and subsets of ). The general form where the phrase eventually (or sufficiently large) is found appears as follows: is eventually true for ( is true for sufficiently large ), where and are the universal and existential quantifiers, which is actually a shorthand for: such that is true 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 some . For example, the definition of a sequence of real numbers converging to some limit is: For each positive number , there exists a natural number such that for all , . When the term "eventually" is used as a shorthand for "there exists a natural number such that for all ", the convergence definition can be restated more simply as: For each positive number , eventually . Here, notice that the set of natural numbers that do not satisfy this property is a finite set; that is, the set is empty or has a maximum element.