Editing Sequence (section)

===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.