3
$\begingroup$

I'm writing some equations dealing with sets and sequences.

I have a sequence $S$ and want to show that $x$ is an element of $S$, however I am hesitant writing $x \in S$ because I don't want to indicate $S$ is a set. I would also prefer not to write 'substring of length one' (e.g. $\hat{x} \subseteq S$, or something to that effect) because x should not be mistaken for a sequence either.

What is the best notation for 'element of a sequence'?

  • 0
    Okay... I should not have said that some people 'never write $(s_k)$'! I will write it like that, using $S = (s_k)_{k \in \mathbb{N}}$ when necessary.2012-08-23

3 Answers 3

1

Remember that a sequence is just a function $S:\mathbb N \rightarrow X$ (with $X$ usually being $\mathbb R$ or $\mathbb N$).

So the set of members of the sequence is just the image of $S$, which can be written as $S(\mathbb N)$ or sometimes $Im(S)$.

Then $x\in S(\mathbb N)$ is a clear and concise way to describe that $x$ appears in $S$.

  • 0
    Or: $\exists n\;(x=S(n)).$2017-05-27
0

I would suggest to use a notation, say ||S||, for the set interpretation of a sequence S.

  • 0
    While it is a drawback, it can still be beneficial if it makes the text more clear.2018-10-25
0

One can use any notation, as long as one introduces it explicitly and chooses it well. A sequence is just an ordered set (and usually an at most countable one), so my advice would be to write "In what follows, I write $x \in S$ to denote that $x$ is an element of the sequence $S$".