Talk:Sequence

Formal definition

The first two sentences of the "Formal definition" section contradict each other. Then in the same paragraph it claims that "interval of integers" rules out finite sequences, which is nonsense. Actually any "formal definition" that eliminates finite sequences is in conflict with vast usage across all of mathematics, even though it is fair to note that sometimes "sequence" is defined to exclude them. McKay (talk) 06:09, 8 January 2026 (UTC)

I agree that section Sequence#Definition needs some cleanup. The problem is that we have three types of sequences (finite, infinite, bi-infinite), and different authors admit different types. Moreover, some properties (like increasing) apply to all types, others (like bounded) don't make sense for finite sequences, etc. Therefore, I have problems to suggest a good overall article structure. I thought about outsourcing finite sequences to the Tuple article (and omitting bi-infinite sequences), but this would e.g. duplicate the increasing definition. - Jochen Burghardt (talk) 10:30, 8 January 2026 (UTC)

Surely the correct thing is to say that formally a sequence is a function from some interval of the integers (allowing four order-types, and indexing in various ways). I agree with McKay that the way the paragraph dwells on the idea that maybe sequences means only N-indexed sequences is odd. I'm going to trim some of that now (not that that will solve any larger issues). --JBL (talk) 20:28, 8 January 2026 (UTC)

How does it look now? --JBL (talk) 20:32, 8 January 2026 (UTC)

Clearly better. I'd add "(finite or infinite)" before "interval" since the target article prominently handles the restricted (finite-only) notion of intervals. And maybe "mapping ... to" is a more common phrase than "function ... to"? (Being a non-native English speaker, I don't dare to have a firm opinion here.) - Jochen Burghardt (talk) 08:33, 9 January 2026 (UTC)

I included the finite or infinite explicitly in the next sentence to avoid two parentheticals in the same sentence. I agree that "function ... to" is a bit awkward, but I think the word "function" itself is slightly less jargony than "mapping"; I think more improvements are possible here but I didn't immediately see what they should be. --JBL (talk) 17:37, 9 January 2026 (UTC)

The paragraph still had a lot of parentheticals, so I tried rephrasing it a bit. Stepwise Continuous Dysfunction (talk) 23:23, 9 January 2026 (UTC)

Does the function ${\displaystyle 1\mapsto 10,2\mapsto 20,3\mapsto 30}$ define the same sequence ${\displaystyle (10,20,30)}$ as the function ${\displaystyle 2\mapsto 10,3\mapsto 20,4\mapsto 30}$, or is it a different sequence that just happens to look the same in standard notation? Also, does ${\displaystyle 2\mapsto 10,4\mapsto 20,6\mapsto 30}$ not define a sequence at all, given that ${\displaystyle [2,4,6]}$ is not an interval? I agree that the function method is a good approach to a formal definition, but I'm not satisfied that it is correctly done. As well as saying what a sequence is, it is necessary to say when two sequences are the same as the equality relation is central to formal applications. Restricting "interval" to the minimum number of types necessary would be one way to try. McKay (talk) 04:08, 27 January 2026 (UTC)

I don't think admitting arbitrary intervals has any advantages. Instead, I suggest to admit the natural numbers (infinite sequence) or an initial segment of them (finite sequence) as domains. This is what e.g. Halmos does in Ch.11 "Numbers" of his book "Naive Set Theory". Bi-infinite sequences are very rare, afaik, and I suggest to outsource them to Net (topology), which is linked under "See also" / "Related concepts". Jochen Burghardt (talk) 17:06, 27 January 2026 (UTC)

Bi-infinite sequences are not rare and not limited to topology: every class in which I've ever introduced the Fibonacci numbers has at some point done a homework exercise in which they discover that the Fibonacci numbers are a bi-infinite sequence, and sequences that are polynomial functions of a positive integer argument but that also have a meaningful interpretation when extended to negative integer arguments are commonplace in combinatorics (see e.g. Erhart polynomial#Reciprocity property, though it seems we lack an article on the general phenomenon of combinatorial reciprocity, at least by that name). Likewise every combinatorial sequence knows who its indices are, with shifted sequences being unquestionably different. --JBL (talk) 17:31, 27 January 2026 (UTC)

Or consider e.g. the coefficients of a Laurent series. –jacobolus (t) 17:41, 27 January 2026 (UTC)

To me both of these questions have answers that I consider immediate and uncontroversial: Does the function ${\displaystyle 1\mapsto 10,2\mapsto 20,3\mapsto 30}$ define the same sequence ${\displaystyle (10,20,30)}$ as the function ${\displaystyle 2\mapsto 10,3\mapsto 20,4\mapsto 30}$, or is it a different sequence that just happens to look the same in standard notation? These are certainly different sequences; they have different generating functions, for example. Also, does ${\displaystyle 2\mapsto 10,4\mapsto 20,6\mapsto 30}$ not define a sequence at all, given that ${\displaystyle [2,4,6]}$ is not an interval? It is a function that is closely related to several sequences (as a bisection, or as an index padding) but it is not itself a sequence. --JBL (talk) 17:25, 27 January 2026 (UTC)

When you say "same sequence" do you mean in some philosophical sense? That seems beyond the scope of this article. If by "same" you mean mathematical equality, then that can be straight-forwardly answered based on whichever definition of 'sequence' you establish. By most definitions, these sequences not equal. But you could certainly define an equivalence relation (where two sequences which re-index to each-other are equivalent) for which they are in the same equivalence class; you could consider the equivalence classes to be the "real" sequences, and the formally defined sequences with specified indices to be a kind of arbitrary choice. –jacobolus (t) 17:36, 27 January 2026 (UTC)

To JBL: I am questioning the definition, not its application. It is bizarre and unnecessary to suppose that every sequence has sitting unseen behind it a particular function that defines it and that (10,20,30) sometimes has one function and sometimes has a different function. To jacobolous: the concept of equality is part of almost every formal system in mathematics. Very few mathematicians would agree that there are a multitude of sequences that are all written as (10,20,30) and reject the statement ${\displaystyle (10,20,30)=(10,20,30)}$ until it is shown that they are defined using the same integer intervals. McKay (talk) 05:16, 12 February 2026 (UTC)

I understand what you're questioning, I just don't think you're right. Mathematicians deal all the time with both labeled and unlabeled objects (i.e., between indexed things and equivalence classes of indexed things) and pass between them without much in the way of note or comment when it doesn't matter. Do you consider it "bizarre and unnecessary" to believe that the unique unlabeled tree with three vertices is an equivalence class of trees? Because in my experience this is the only way that one ever defines unlabeled trees. Likewise the notation (10, 20, 30) is for an equivalence class of sequences. Just as people happily use unlabeled trees when the identities of the vertices don't matter, people happily use unlabeled sequences in contexts when the precise indices are irrelevant. But when you actually pin someone down on what the unlabeled object "really is", you either get an equivalence class or a fixed representative of the class. --JBL (talk) 17:24, 12 February 2026 (UTC)

But who says (10,20,30) is an equivalence class of sequences? Where is the source? You can't just apply the definition I am questioning to disprove the objection; that is circular reasoning. McKay (talk) 06:50, 2 March 2026 (UTC)

The notation (10, 20, 30) is something that you have introduced into this conversation, claiming that it is something called "the standard notation" of a finite sequence. It is possible that you personally have in your head a definition of "the standard notation of a finite sequence" that conflicts with the (well attested in reliable sources) definition of sequences as functions, and also that somehow is not just what everyone else means by this (which, again, is one of two things, either an equivalence class or a fixed (though context-dependent) representative of the equivalence class). But I don't see how the mismatch between your personal expectations and the definition in reliable sources is a problem that I need to fix by finding a source. --JBL (talk) 17:22, 2 March 2026 (UTC)

Please show us a source that does not use notation like (10,20,30). In searching for an hour I found no differences other than choice of punctuation, ergo calling it standard notation is perfectly correct. Meanwhile, in the very few sources which provide a formal definition, I found finite sequences defined as functions with domain ${\displaystyle 1,2,...,n}$ or as functions with domain ${\displaystyle 0,1,...,n}$ (eg. Halmos) but never both options at the same time. As far as I know, every author who defines finite sequences as functions chooses one particular interval of integers as the domain and (10,20,30) corresponds to a single function, either ${\displaystyle 1\mapsto 10,2\mapsto 20,3\mapsto 30}$ or ${\displaystyle 0\mapsto 10,1\mapsto 20,2\mapsto 30}$ depending on the author's preference. If you want us to allow arbitrary finite intervals, yes you do need to provide a source. McKay (talk) 03:50, 4 March 2026 (UTC)

Ignoring the wrapping of the first two and final sentence, the substantive part of this message in in complete agreement with the statement that when the indices don't matter, people make a choice of a fixed (though context-dependent) representative of the equivalence class. --JBL (talk) 17:20, 4 March 2026 (UTC)

Not true. It is the people who write the formal definition who make the choice. Those who adopt the definition no longer have a choice. If I adopt Halmos' definition, all my sequences that have ${\displaystyle n}$ entries are functions with domain ${\displaystyle \{0,1,\ldots ,n-1\}}$. McKay (talk) 03:53, 9 March 2026 (UTC)

It is possible that you don't understand me; it is possible that is partly my fault; but it is certainly the case that your unfounded self-confidence in your assertion that a very simple and self evident statement is "not true" has sapped me of any further desire to reach a mutual understanding with you. --JBL (talk) 17:26, 9 March 2026 (UTC)

I think the fundamental conflict here is between people who believe words' meaning is primarily determined by common practice vs. people who believe words' meaning is primarily determined by formal definitions. This is fundamentally the same as the conflict between descriptivists and prescriptivists in language use more generally. Mathematics has the additional wrinkle that (1) fully specified and completely rigorous formal proofs (e.g. ones you want to verify with a computer program) must always have an exact definition given for each word used, but (2) proofs are usually written with many details elided and irrelevancies ignored for the sake of better communicating to an expected human audience, and usually the precise differences between possible definitions of terms do not matter to the substance of the argument.

In practice mathematicians usually adopt whichever meaning is most convenient for their context, commonly leaving terms undefined (or sometimes explicitly adopt one specific definition but then implicitly switch to one that is different where it is convenient), expecting readers to ignore (or fill in for themselves) the trivial details of converting between variant formal definitions wherever necessary. –jacobolus (t) 18:43, 9 March 2026 (UTC)

Incidentally, I assumed until now that the "formal definition" was sourced to the book of Gaughan cited at the end of that paragraph. However it isn't. Gaughan only defines one-way infinite sequences, as functions with the natural numbers as domain. So it seems we have no source at all. This is an obvious no-no. To (almost) follow Jochen, I propose we allow as domains only all the integers, the natural numbers, or an initial interval of natural numbers. Then each sequence has a unique function and equality of the functions is a perfect match to the everyday meaning of equality of sequences. McKay (talk) 05:16, 12 February 2026 (UTC)

This leaves us still without a source. The only one I have at hand is Halmos (see above).

If JayBeeEll and jacobolus still insist on bi-infinite sequences to be subsumed here, they should provide one of their sources. I guess this will at least introduce another case distinction for the domain (natural numbers, initial segment, integer numbers), or even the above quotient set construction by jacobolus (avoiding case distinctions, though).

That was my motivation to suggest outsourcing bi-infinite sequences. If Net (topology) is not acceptable, then what about an own section (called e.g. "Generalizations") at the end of Sequence? - Jochen Burghardt (talk) 10:17, 12 February 2026 (UTC)

There are thousands of sources using this concept (in a wide variety of different topics and contexts), which aren't too hard to find by keyword search. I see lots of examples like "Let ⁠${\displaystyle (a_{n})_{n\in \mathbb {Z} }}$⁠ be a sequence ...", but it's hard to find a source focused specifically on defining the concept of integer-indexed sequences. I'm sure with diligent search someone could find a reasonable citation to point readers toward, but I'm not finding a good one with just a few minutes of searching. –jacobolus (t) 16:27, 12 February 2026 (UTC)

To comment on the question of why Net (topology) is not acceptable: consider that already the first section Fibonacci_sequence#Definition of the article on the Fibonacci sequence introduces the idea that it (or can be extended to be) a bi-infinite sequence, but surely there is no question that a link to Net (topology) would be inappropriate there. When I use a standard search engine for the words "bi-infinite sequence", here are two of the top hits, both of which seem like viable sources for some portion of the content in question: . --JBL (talk) 19:29, 12 February 2026 (UTC)

Source [1] seems to define only infinite sequences, and source [2] only bi-infinite sequences, or did I miss something? - Jochen Burghardt (talk) 11:00, 13 February 2026 (UTC)

In the first reference (which is apparently Topological and Ergodic Theory of Symbolic Dynamics by Henk Bruin), the sentence "We are interested in ..." that starts on the bottom of page 1 and finishes on page 2 provides a definition of both infinite and bi-infinite sequences. --JBL (talk) 17:53, 13 February 2026 (UTC)

That's my point. We'd need a definition that works for finite sequences also. - Jochen Burghardt (talk) 17:17, 14 February 2026 (UTC)

I am not convinced that we need a single reference that says that the word "sequence" simultaneously means all these different things, when it is an uncontentious fact that it does mean that -- it should suffice to have good-quality references that attest to the different notions. (Compare the (much worse) situation at Polyhedron#Definition.) This is particularly the case in mathematics, where each author will likely choose "sequence" to have one meaning and then use other related phrases ("finite sequence", "word", etc.) when separating meanings is important. For example, Bruin certainly contemplates finite sequences, it's just that he calls them "words" or "blocks" or "factors". (And then you can see that occasionally "word" is also used for an infinite object, as in Example 1.6.) --JBL (talk) 18:20, 14 February 2026 (UTC)

It's also not entirely necessary that our initial description or definition of 'sequence' is general enough to cover every use of the word 'sequence' anywhere in mathematics. We can start with a definition found most commonly in sources, and then provide alternatives or generalizations later down the page. –jacobolus (t) 18:23, 14 February 2026 (UTC)

The best way to make progress on this is to separate out the three types (finite, one-way infinite, two-way infinite), and give a definition for each. It seems hopeless to find a source that defines all of them at once. For finite sequence, we can cite Halmos for a function with domain an interval of integers starting at 0. If we have a source that starts the domain at 1, we can mention that as well. For 1-way infinite sequences, there are plenty of sources that use the natural numbers as domain, with the same variation of starting at 0 or 1. In both cases, we can't (by WP:NOR) allow arbitrary starting points without a reliable source that allows them. For 2-way infinite sequences, I've only ever seen all of the integers used as the domain. McKay (talk) 04:06, 4 March 2026 (UTC)

Sounds fine to me (cf. my above suggestion of 12 Feb, to move bi-infinite sequences to a section "Generalizations"). - Jochen Burghardt (talk) 10:25, 4 March 2026 (UTC)

We do not need a source that defines all of them at once to discuss them all together; the situation at Polyhedron is relevant. For two-way infinite sequences, everyone in this discussion agrees that those words mean that the domain is ${\displaystyle \mathbb {Z} }$. For one-way infinite sequences, there are examples above (the sequence of coefficients of a Laurent series) that present a serious problem for the idea that people only ever use the domain ${\displaystyle \mathbb {N} }$ (in one of its two meanings). It is unfortunate that most sources take "sequence" as a primitive/intuitive/understood notion rather than write down a careful definition, but it is also not difficult to see that people in practice use the word "sequence" to mean something more flexible than you're trying to allow here.

More broadly I think your argument proves too much. If there is no source that says that a (finite) sequence is a either a function with domain {0, 1, ..., n} or a function with domain {1, 2, ..., n}, your arguments suggest it is OR for us to say that. (That is, not a source that supports one of these, but a source that supports that the definition of sequence is the disjunction.) To be clear, I do not accept this conclusion; but I think your argument leads to a conclusion that forbids your proposed solution. Personally I would be very comfortable with an assertion that sequences most commonly are indexed starting at 0 or 1, with our without a source that says that explicitly. --JBL (talk) 17:33, 4 March 2026 (UTC)

Any number of indexing functions can be specified for a sequence, and the domains don't have to be intervals, or integers, or in increasing order, or even from an ordered set. That is quite distinct from the unique function used to define the sequence. To formally define an indexing function one chooses a map from the index set to the domain of the defining function; it is an additional step beyond just defining the sequence. McKay (talk) 04:28, 9 March 2026 (UTC)

It occurred to me this afternoon to check whether the OEIS defines the term "sequence". It does not. However, each sequence in the OEIS contains an offset, which they define as "the index of the first term". This is not usable as a reference for anything obviously, but it is another piece of evidence in favor of the idea that sequences are not necessarily 0- or 1-indexed. --JBL (talk) 21:41, 4 March 2026 (UTC)

The OEIS is the best example you found but it is not a reliable source for formal definitions. The OEIS does not mention the defining function for the sequence, instead the offset defines an index function for it. An offset of 3 specifies an indexing function that maps 3 to first, 4 to second, etc. McKay (talk) 04:28, 9 March 2026 (UTC)

Concrete proposal: A finite sequence of length ${\displaystyle n}$ is a function whose domain is ${\displaystyle \{0,1,\ldots ,n-1\}}$.[Halmos] An infinite sequence is a function whose domain is the natural numbers.[Halmos or someone else]. Some authors also define a two-way infinite sequence, which is a function whose domain is the integers.[Source] In each case, the natural ordering of the integers serves to specify the order in which the elements of the sequence appear: the image of ${\displaystyle m}$ comes earlier than the image of ${\displaystyle n}$ if ${\displaystyle m<n}$.

That is a complete formal definition and it also says how the formal definition maps to the informal definition. I'm going to put this in if nobody makes a policy-based argument against it (which mandates a source). McKay (talk) 04:28, 9 March 2026 (UTC)

Here's an example of a source with a more general definition:

▸ Definition (Sequence of Numbers) A sequence of numbers is a relation between the set of natural numbers ⁠${\displaystyle \mathbb {N} }$⁠ and the set of real numbers ⁠${\displaystyle \mathbb {R} }$⁠ such that with each element (index) of the domain ⁠${\displaystyle n\in \mathbb {N} }$⁠ there is associated a unique number ⁠${\displaystyle a_{n}\in \mathbb {R} }$⁠. Hence, a sequence of numbers is a function whose domain is ⁠${\displaystyle \mathbb {N} }$⁠ and the image is a subset of ⁠${\displaystyle \mathbb {R} }$⁠.

This definition is usually extended to the case of a more general domain ⁠${\displaystyle \mathbb {N} _{0}}$⁠, which represents a subset of the integers ⁠${\displaystyle \mathbb {Z} }$⁠ such that ${\displaystyle \mathbb {N} _{0}={}}$${\displaystyle \{k_{1},k_{2}=k_{1}+1,{}\!}$${\displaystyle k_{3}=k_{2}+1,\ldots ,{}\!}$${\displaystyle k_{i+1}=k_{i}+1,\ldots \}}$, where ⁠${\displaystyle k_{1}\in \mathbb {Z} }$⁠. The point of this extension is to make possible the counting of the sequence elements starting from an arbitrary initial index (⁠${\displaystyle k_{1}}$⁠ instead of ⁠${\displaystyle 1}$⁠) still keeping the structure and order of natural numbers: each next index (the element of the domain) is obtained adding ⁠${\displaystyle 1}$⁠ to the current index. (Evidently, in this extension, the number of negative indices is finite or empty.)

Bourchtein & Bourchtein (2022). "Sequences of Numbers". Theory of Infinite Sequences and Series. Birkhäuser, Cham. doi:10.1007/978-3-030-79431-6_1

(Later, they also define a sequence of functions likewise, again noting the extension to an arbitrary initial index.)

–jacobolus (t) 06:27, 9 March 2026 (UTC)

Here's a different example, from a book that never explicitly formally defines "sequence" but uses it in a way similar to JBL's concept:

6.7 Given sequence ⁠${\displaystyle 1,8,27,64,125,\ldots }$⁠ reads as follows [...] As an exercise, the same sequence may be written with any other initial indices. That is to say, the initial index value may be arbitrary chosen, without changing the total number of terms nor the total sum, as, for example, $${\displaystyle {\begin{aligned}S_{n}&=1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+\cdots \\[3mu]&=\sum _{n=1}^{\infty }n^{3}\\&=\sum _{n=0}^{\infty }(n+1)^{3}\equiv (0+1)^{3}+(1+1)^{3}+(2+1)^{3}+(3+1)^{3}+\cdots \\&=\sum _{n=-1}^{\infty }(n+2)^{3}\equiv (-1+2)^{3}+(0+2)^{3}+(1+2)^{3}+(2+2)^{3}+\cdots \end{aligned}}}$$ or any other arbitrary integer choice of "the first position" index.

6.8. Given sequence ⁠${\displaystyle 1,2,4,8,16,32,\ldots }$⁠, reads as follows: on position ⁠${\displaystyle n=1}$⁠ term ⁠${\displaystyle a=1}$⁠, on position ⁠${\displaystyle n=2}$⁠ term ⁠${\displaystyle a=2}$⁠, on position ⁠${\displaystyle n=3}$⁠ term ⁠${\displaystyle a=4}$⁠, on position ⁠${\displaystyle n=4}$⁠ term ⁠${\displaystyle a=8}$⁠, etc. Using mathematical notation, that statement is written as $${\displaystyle a_{1}=1,\ a_{2}=2,\ a_{3}=4,\ a_{4}=8,\ a_{5}=16,\ \ldots }$$ where "..." implies "and so on until infinity." However, one has to recognize that this sequence is known as "the powers of two", i.e., if ⁠${\displaystyle n=0,1,2,3,\ldots }$⁠, then there is a ⁠${\displaystyle 2^{n}}$⁠ number that correspond to the given sequence. Note that the first term equals ⁠${\displaystyle {1}}$⁠; thus using mathematical notation, the indices are shifted as $${\displaystyle a_{0}=1,\ a_{1}=2,\ a_{2}=4,\ a_{3}=8,\ \ldots }$$ so that powers of two are correlated with ⁠${\displaystyle n}$⁠ as $${\displaystyle a_{n}=2^{n},\quad n=0,1,2,3,\ldots .}$$

Sobot (2025) Engineering Mathematics by Example, vol. 3, doi:10.1007/978-3-031-81104-3 –jacobolus (t) 19:11, 9 March 2026 (UTC)

I strongly oppose McKay's suggestion, which is both factually wrong and a poor reflection of the breadth of reliable sources on the topic; the relevant policy here is WP:CONSENSUS. --JBL (talk) 17:38, 9 March 2026 (UTC)

Could you give (or repeat / refer to? - I lost overview) your suggestion? - Jochen Burghardt (talk) 18:49, 9 March 2026 (UTC)

Sorry if my example distracted from the proposed change. Inre McKay and JBL's disagreement: personally, I think it's fine if we say that a "sequence" is a function from the natural numbers to some set, so long as we also mention that the choice of initial index is arbitrary and is routinely shifted where convenient. I think trying to explicitly define a sequence as an equivalence class of functions with shifted domains would be unnecessarily confusing, and readers for whom that matters can figure it out. –jacobolus (t) 19:15, 9 March 2026 (UTC)

@Jacobolus: I'd directed my question to JayBeeEll. - Jochen Burghardt (talk) 16:29, 10 March 2026 (UTC)

@JayBeeEll: Could you give (or repeat / refer to?) your suggestion? - Jochen Burghardt (talk) 16:29, 10 March 2026 (UTC)

@Jochen Burghardt: Sorry, none of your attempts to ping me worked, so I didn't see this until just now. Personally I am happy with the current section "Definition" (that is a subsection of "Formal definition and basic properties"). If someone wanted to add the assertion that most commonly sequences start with index 0 or 1, I would probably be okay with that, too.

Incidentally, here is an example of a paper that very explicitly treats sequences as functions on (finite) intervals of integers; the literature that touches on a primitive object like this is vast. --JBL (talk) 17:26, 14 March 2026 (UTC)

What if, as a compromise, we say something along the lines of "A sequence is usually defined as ..." or "A sequence can be defined as ...", giving a definition where we enumerate the elements starting from (e.g.) 1, but then explicitly mention that the starting index is arbitrary and where convenient many authors (explicitly or implicitly) use sequences starting from some other integer. McKay does seem correct in saying that the most common formal definition found in reliable sources (especially introductory or broad-focused sources) is of this type.

One way or the other, we should probably include a section explicitly talking about shifting (re-indexing) a sequence, and perhaps also more information about operations like convolution. –jacobolus (t) 17:47, 14 March 2026 (UTC)

Perhaps "A sequence can be defined as a function whose domain is a subset of the natural numbers. The choice of the smallest element of the domain, the so-called starting index, is fundamentally arbitrary but is often either 0 or 1, depending on what is convenient for the purpose at hand." Stepwise Continuous Dysfunction (talk) 20:13, 14 March 2026 (UTC)