Eventually (mathematics)

In the mathematical areas of number theory and analysis, an infinite sequence (a_n) is said eventually to have a certain property if all terms beyond some (finite) point in the sequence have that property. 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 does not 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 (a_n) converging to some limit a is: for all ε > 0 there exists N > 0 such that, for all n > N, |a_n − 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 |a_n − 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.

Notation

The phrase eventually (or sufficiently large) is used in such contexts as:

P

is eventually true for

x

/ :

P

is true for sufficiently large

x

which is actually shorthand for:

\exists ~a\in {\mathbb {R}}

such that

P

is true

\forall ~x\geq a

or, somewhat more formally:

\exists~ a\in\mathbb{R}:\forall~ x\in\mathbb{R}:x\ge a \Rightarrow P(x)

This does not necessarily mean that any particular value for $a$ is known, but only that such an $a$ exists. The phrase "sufficiently large" should not be confused with the phrases "arbitrarily large" or "infinitely large".

Other uses in mathematics

A 3-manifold is called sufficiently large if it contains a properly embedded 2-sided incompressible surface. This property is the main requirement for a 3-manifold to be called a Haken manifold.

Examples

All primes above 2 are odd can be written as "Eventually, all primes are odd"

In the long run, all primes are congruent to ±1 mod 6

The square of a prime is congruent to 1 mod 24, given that the prime is above 3

This can be generalized as $p^{n}\equiv 1{\pmod {x}}$ A conjecture can then be made for the closed form function $f(n)=x$ as well as the minimum p.

References

Weisstein, Eric W. "Sufficiently Large". MathWorld. Margherita Barile. "Eventually". MathWorld.

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