Are sequences properly denoted as $\subset$ of a set, or $\in$ a set?

Question

Given some sequence $(x_n)$ of some subset $M \subset \mathbb{R^n}$, is it more appropriately to denote $(x_n) \subset M$, or $ (x_n)\in M$?

This stems from confusion of using "in" i.e. whenever people say "suppose $(x_n)$ is a sequence in $M$.

Answer 1

Neither $(x_n)\subset M$ nor $(x_n)\in M$ should be used, since neither is correct. The fact that we say "sequence in $M$" does not mean we should use the symbol for element in a set. There are clear and precise definition for the meaning of $\in $ and $\subset$ and they dictate what the correct usage is. Generally speaking, a sequence $(x_n)$ of elements in $M$ is not itself an element of $M$, and so $(x_n)\in M$ is not justified. Further, $(x_n)$ is a sequence, not a set, and thus $\subset$ certainly isn't justified. There is no accepted symbol to denote a sequence in a set $M$, though formally speaking a sequence in $M$ is precisely a function $\mathbb N \to M$. The set of all these functions is commonly denoted by $M^{\mathbb N}$, so writing $(x_n)\in M^{\mathbb N}$ is equivalent to the claim that $(x_n)$ is a sequence of elements in $M$. So technically, there is existing notation, but it is not often used.

Answer 2

Formally speaking, neither is correct. Sequences are functions, and the elements of $M$ are not [usually] functions from $\Bbb N$ to $M$ itself (although that is known to be possible).

What is correct is that "$(x_n)$ is a sequence such that $x_n\in M$ [for all $n$]".

To me, then, both notations abuse about the same the correct way of phrasing, so both are acceptable. The important thing is that people understand what you meant. But if you want to be safe, open a couple of books in the field in which you want to write this, and see how they write it.