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125:, if it does not 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
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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
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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
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introduces an operator that can be used to express statements interpretable as: Certain property will eventually hold in a future moment in time.
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of a natural number eventually ends in the digit 0 (specifically, this is true for all natural numbers greater than 4).
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of a prime is eventually congruent to 1 mod 24 (specifically, this is true for all primes greater than 3).
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terms" (although "almost all" can also be used to allow for infinitely many exceptions as well).
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At the basic level, a sequence can be thought of as a function with natural numbers as its
470:{\displaystyle \exists a\in \mathbb {R} :\forall x\in \mathbb {R} :x\geq a\Rightarrow P(x)}
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of natural numbers that do not satisfy this property is a finite set; that is, the set is
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exists. The phrase "sufficiently large" should not be confused with the phrases "
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Arbitrarily large#Arbitrarily large vs. sufficiently large vs. infinitely large
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has the said property. This notion is used, for example, in the study of
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1246:. This property is the main requirement for a 3-manifold to be called a
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is called sufficiently large if it contains a properly embedded 2-sided
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is said to have some property eventually if there is an element
545:, or equivalently, that the property is satisfied by one of its
934:{\displaystyle \left\vert a_{n}-a\right\vert <\varepsilon }
862:", the convergence definition can be restated more simply as:
801:{\displaystyle \left\vert a_{n}-a\right\vert <\varepsilon }
26:
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This does not necessarily mean that any particular value for
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is used as a shorthand for "there exists a natural number
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54:but its sources remain unclear because it lacks
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340:{\displaystyle \exists a\in \mathbb {R} }
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85:Learn how and when to remove this message
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313:, which is actually a shorthand for:
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163:The general form where the phrase
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885:{\displaystyle \varepsilon >0}
615:{\displaystyle N\in \mathbb {N} }
587:{\displaystyle (a_{n})_{n\geq N}}
500:is known, but only that such an
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389:{\displaystyle \forall x\geq a}
171:) is found appears as follows:
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399:or somewhat more formally:
703:{\displaystyle \varepsilon }
148:{\displaystyle \mathbb {R} }
1210:Eventually, all primes are
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1149:{\displaystyle x>x_{0}}
1233:Other uses in mathematics
866:For each positive number
812:When the term "eventually
690:For each positive number
536:Motivation and definition
1361:Mathematical terminology
298:{\displaystyle \exists }
278:{\displaystyle \forall }
40:This article includes a
1035:such that the function
957:of the expression "for
654:{\displaystyle (a_{n})}
311:existential quantifiers
129:(such as sequences and
69:more precise citations.
1244:incompressible surface
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855:{\displaystyle n>N}
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1302:"Sufficiently Large"
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661:converging to some
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42:list of references
18:Sufficiently large
1088:{\displaystyle f}
1068:{\displaystyle s}
1048:{\displaystyle f}
1028:{\displaystyle S}
1008:{\displaystyle s}
988:{\displaystyle S}
829:{\displaystyle N}
726:{\displaystyle N}
710:, there exists a
678:{\displaystyle a}
522:arbitrarily large
513:{\displaystyle a}
493:{\displaystyle a}
360:{\displaystyle P}
254:{\displaystyle x}
231:{\displaystyle P}
211:{\displaystyle x}
187:{\displaystyle P}
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1334:. Retrieved
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1327:"Eventually"
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1309:. Retrieved
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1188:Hardy fields
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974:
963:
955:special case
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627:real numbers
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547:subsequences
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238:is true for
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99:mathematical
96:
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61:Please help
53:
1366:3-manifolds
594:, for some
127:ordered set
117:is said to
67:introducing
1350:Categories
1336:2019-11-20
1311:2019-11-20
1288:References
1267:Almost all
1240:3-manifold
959:almost all
543:eventually
526:infinitely
347:such that
196:eventually
165:eventually
119:eventually
1226:factorial
1212:congruent
929:ε
915:−
874:ε
796:ε
782:−
698:ε
605:∈
577:≥
453:⇒
447:≥
433:∈
427:∀
416:∈
410:∃
381:≥
375:∀
330:∈
324:∃
307:universal
293:∃
273:∀
198:true for
101:areas of
75:July 2018
1261:See also
1194:Examples
367:is true
305:are the
159:Notation
123:property
115:function
111:sequence
107:analysis
1075:, then
131:subsets
97:In the
63:improve
1219:square
1201:primes
966:domain
524:" or "
265:where
1199:"All
951:empty
663:limit
113:or a
48:, or
1224:The
1217:The
1134:>
926:<
877:>
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793:<
744:>
686:is:
309:and
285:and
167:(or
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105:and
1205:odd
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194:is
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