Definition talk:Term of Sequence

Viewing a sequence as a mapping. Is a term an element of the range of the mapping or an element of the graph of the mapping?

I believe it is used in both senses. To demonstrate consider the sequence: $\langle 2, 11, 5, 2, 3 \rangle$

$(1)$ We say that $2$ is a term of the sequence. (In this case we are pointing to $2$ in the range of the sequence)

$(2)$ We say that $2$ is the $4^{th}$ term of the sequence. (In this case we are pointing to $\left({4, 2}\right)$ in the graph of the sequence).

Is this correct and if so should we point it out on the Definition page?

I am sorry to post such persnickety things (I just want things to be absolutely clear). --Jshflynn 09:08, 17 August 2012 (UTC)


 * Correct. I would be inclined to say that phrasing $(1)$ would also deserve a position on the def page. --Lord_Farin 09:15, 17 August 2012 (UTC)


 * I see where you're coming from: "an element of a sequence" is far too imprecise, unless you go with the first (non-rigorous definition) "A sequence is a set of objects listed in a specific order." My view: put the imprecise intuitive definition later in the page and make the rigorous definition, that a sequence is a mapping from a subset of $\N$ to a set $S$, the first thing the reader sees. Then a "term" would be "an element in the range of a sequence" with a possible "also defined as" as "an element in the graph of a sequence".--prime mover 12:33, 17 August 2012 (UTC)

Let's cut the crap. It's not helpful at all to consider terms as elements of the range of a sequence, and every sense to consider them as elements of the graph. If you just want to consider the range, then consider the range as a set and treat it as a set - but a term of a sequence has its preimage implicit.

Thus the concept of an "infinite sequence" (which currently looks a bit silly) can be defined as appropriate: "a sequence with an infinite number of terms" rather than the circumlocutions that are currently needed.

Can someone find a reliable definition and cite it? --prime mover (talk) 22:50, 30 October 2012 (UTC)


 * I have here "Foundations of Higher Mathematics - Third Edition" by Fletcher and Wayne Patty. Op.cit., p. 154: "The $n$th term of a sequence $\left\langle{f \left({n}\right)}\right\rangle$ is $\left({n, f \left({n}\right)}\right)$.", i.e. precisely what we want. --Lord_Farin (talk) 23:06, 30 October 2012 (UTC)
 * BTW, when time comes, I may start covering cited work, for it (quote from preface) "...introduces students to basic techniques of writing proofs and acquaints them with some fundamental ideas that are used throughout mathematics." --Lord_Farin (talk) 23:10, 30 October 2012 (UTC)


 * Thanks for tending to this guys. I am glad the first sense was abandoned. --Jshflynn (talk) 17:39, 31 October 2012 (UTC)


 * I've put the symbolism together - it needs the words and links added appropriately so as to describe it as the element of the ordered pair - it's late and I need sleep. --prime mover (talk) 23:11, 30 October 2012 (UTC)


 * Okay, then; there we go. Now, is the title of Definition:Sequence of Distinct Terms compatible with that, or should we change its name? (By the way, I'd just call it an injective sequence.) Or does it not matter? --abcxyz (talk) 23:12, 30 October 2012 (UTC)


 * You're dead right, Definition:Sequence of Distinct Terms is completely incompatible with this. The whole paradigm needs a complete rewrite, starting with the definition of Mapping. Or Relation. Or Set. or Object. I don't know, I've got so tied in the knots of exhausting trivialities I've lost the ability to think. --prime mover (talk) 21:10, 31 October 2012 (UTC)


 * I'm assuming that we should also change Definition:Term of Family accordingly, yes? --abcxyz (talk) 23:13, 30 October 2012 (UTC)

Stylistic pedantry
I note abcxyz replacing "$k$th" by "$k$th". The former is almost universal on ProofWiki, and I'm favouring it for reasons of clarity and reproducibility. I think it's good to cast a vote on which of these should prevail, so as to avoid unnecessary work, reverting and irritations. --Lord_Farin (talk) 13:51, 31 October 2012 (UTC)


 * I didn't know that the former is almost always used on this website. By the way, could you please explain what "reproducibility" you are talking about? --abcxyz (talk) 14:10, 31 October 2012 (UTC)


 * It's tedious to use &lt;sup&gt; all the time. I may not have chosen the right word - do you know an adverb corresponding to "easily reproducible"? "easily-reproducibility" doesn't count. --Lord_Farin (talk) 14:15, 31 October 2012 (UTC)


 * Anything which avoids us having to use html tags is a good thing. I'm with L_F here: "$k$th" is oldfashioned and fiddly to maintain.