Editing Sequence (section)

==Formal definition and basic properties==
There are many different notions of sequences in mathematics, some of which (''e.g.'', [[exact sequence]]) are not covered by the definitions and notations introduced below.

===Definition===
In this article, a sequence is formally defined as a [[function (mathematics)|function]] whose [[domain of a function|domain]] is an [[Interval (mathematics)|interval]] of [[integers]]. This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). However, many authors use a narrower definition by requiring the domain of a sequence to be the set of [[natural numbers]]. This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition. In some contexts, to shorten exposition, the [[codomain]] of the sequence is fixed by context, for example by requiring it to be the set '''R''' of real numbers,<ref name="Gaughan" /> the set '''C''' of complex numbers,<ref name=Saff>{{Cite book |title=Fundamentals of Complex Analysis |chapter=Chapter 2.1 |chapter-url=https://books.google.com/books?id=fVsZAQAAIAAJ&q=saff+%26+Snider |author=Edward B. Saff & Arthur David Snider |year=2003 |publisher=Prentice Hall |isbn=978-01-390-7874-3 |access-date=2015-11-15 |archive-date=2023-03-23 |archive-url=https://web.archive.org/web/20230323163811/https://books.google.com/books?id=fVsZAQAAIAAJ&q=saff+%26+Snider |url-status=live }}</ref> or a [[topological space]].<ref name=Munkres>{{Cite book|title=Topology|chapter=Chapters 1&2|chapter-url=https://books.google.com/books?id=XjoZAQAAIAAJ|author=James R. Munkres|isbn=978-01-318-1629-9|year=2000|publisher=Prentice Hall, Incorporated |access-date=2015-11-15|archive-date=2023-03-23|archive-url=https://web.archive.org/web/20230323163811/https://books.google.com/books?id=XjoZAQAAIAAJ|url-status=live}}</ref>

Although sequences are a type of function, they are usually distinguished notationally from functions in that the input is written as a subscript rather than in parentheses, that is, {{math|''a<sub>n</sub>''}} rather than {{math|''a''(''n'')}}. There are terminological differences as well: the value of a sequence at the lowest input (often 1) is called the "first element" of the sequence, the value at the second smallest input (often 2) is called the "second element", etc. Also, while a function abstracted from its input is usually denoted by a single letter, e.g. ''f'', a sequence abstracted from its input is usually written by a notation such as <math display=inline>(a_n)_{n\in A}</math>, or just as <math display=inline>(a_n).</math> Here {{math|''A''}} is the domain, or index set, of the sequence.

Sequences and their limits (see below) are important concepts for studying topological spaces. An important generalization of sequences is the concept of [[net (mathematics)|nets]]. A '''net''' is a function from a (possibly [[uncountable]]) [[directed set]] to a topological space. The notational conventions for sequences normally apply to nets as well.

===Finite and infinite===
{{See also|ω-language}}
The '''length''' of a sequence is defined as the number of terms in the sequence.

A sequence of a finite length ''n'' is also called an [[n-tuple|''n''-tuple]]. Finite sequences include the '''empty sequence'''&nbsp;(&nbsp;) that has no elements.

{{anchor|Doubly infinite|Doubly-infinite sequences}}
Normally, the term ''infinite sequence'' refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element. Such a sequence is called a '''singly infinite sequence''' or a '''one-sided infinite sequence''' when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a '''bi-infinite sequence''', '''two-way infinite sequence''', or '''doubly infinite sequence'''. A function from the set '''Z''' of ''all'' [[integers]] into a set, such as for instance the sequence of all even integers ( ..., −4, −2, 0, 2, 4, 6, 8, ... ), is bi-infinite. This sequence could be denoted <math display=inline>{(2n)}_{n=-\infty}^{\infty}</math>.

===Increasing and decreasing===
A sequence is said to be ''monotonically increasing'' if each term is greater than or equal to the one before it. For example, the sequence <math display=inline>{(a_n)}_{n=1}^{\infty} </math> is monotonically increasing if and only if <math display=inline>a_{n+1} \geq a_n</math> for all <math>n \in \mathbf N.</math> If each consecutive term is strictly greater than (>) the previous term then the sequence is called '''strictly monotonically increasing'''. A sequence is '''monotonically decreasing''' if each consecutive term is less than or equal to the previous one, and is '''strictly monotonically decreasing''' if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a '''monotone''' sequence. This is a special case of the more general notion of a [[monotonic function]].

The terms '''nondecreasing''' and '''nonincreasing''' are often used in place of ''increasing'' and ''decreasing'' in order to avoid any possible confusion with ''strictly increasing'' and ''strictly decreasing'', respectively.

===Bounded===
If the sequence of real numbers (''a<sub>n</sub>'') is such that all the terms are less than some real number ''M'', then the sequence is said to be '''bounded from above'''. In other words, this means that there exists ''M'' such that for all ''n'', ''a<sub>n</sub>'' ≤ ''M''. Any such ''M'' is called an ''upper bound''. Likewise, if, for some real ''m'', ''a<sub>n</sub>'' ≥ ''m'' for all ''n'' greater than some ''N'', then the sequence is '''bounded from below''' and any such ''m'' is called a ''lower bound''. If a sequence is both bounded from above and bounded from below, then the sequence is said to be '''bounded'''.

===Subsequences===
A '''[[subsequence]]''' of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of the positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved.

Formally, a subsequence of the sequence <math>(a_n)_{n\in\mathbb N}</math> is any sequence of the form <math display=inline>(a_{n_k})_{k\in\mathbb N}</math>, where <math>(n_k)_{k\in\mathbb N}</math> is a strictly increasing sequence of positive integers.

===Other types of sequences===
Some other types of sequences that are easy to define include:
* An '''[[integer sequence]]''' is a sequence whose terms are integers.
* A '''[[polynomial sequence]]''' is a sequence whose terms are polynomials.
* A positive integer sequence is sometimes called '''multiplicative''', if ''a''<sub>''nm''</sub> = ''a''<sub>''n''</sub> ''a''<sub>''m''</sub> for all pairs ''n'', ''m'' such that ''n'' and ''m'' are [[coprime]].<ref>{{cite book|title=Lectures on generating functions|last=Lando|first=Sergei K.|publisher=AMS|isbn=978-0-8218-3481-7|chapter=7.4 Multiplicative sequences|date=2003-10-21}}</ref> In other instances, sequences are often called ''multiplicative'', if ''a''<sub>''n''</sub> = ''na''<sub>1</sub> for all ''n''. Moreover, a ''multiplicative'' Fibonacci sequence<ref>{{cite journal|title=Fibonacci's multiplicative sequence|first=Sergio|last=Falcon|journal=International Journal of Mathematical Education in Science and Technology|volume=34|issue=2|pages=310–315|doi=10.1080/0020739031000158362|year = 2003|s2cid=121280842}}</ref> satisfies the recursion relation ''a''<sub>''n''</sub> = ''a''<sub>''n''−1</sub> ''a''<sub>''n''−2</sub>.
* A [[binary sequence]] is a sequence whose terms have one of two discrete values, e.g. [[base 2]] values (0,1,1,0, ...), a series of coin tosses (Heads/Tails) H,T,H,H,T, ..., the answers to a set of True or False questions (T, F, T, T, ...), and so on.