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 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 (such as sequences and subsets of formula_1). The general form where the phrase eventually (or sufficiently large) 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 formula_10 is known, but only that such an formula_10 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 formula_12 converging to some limit "formula_10" is:  When the term "eventually""" is used as a shorthand for "there exists a positive number formula_15 such that for all formula_19", 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" (although "almost all" can also be used to allow for infinitely many exceptions as well). 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 formula_22 is such a set and there is an element formula_23 in formula_22 such that the function formula_25 is defined for all elements greater than formula_23, then formula_25 is said to have some property eventually if there is an element formula_28 such that whenever "formula_29", formula_30 has the said property.