How to denote an arbitrary expression involving some number of dummy variables?

Question

Let me give you an example.

$(n)_{n=0}^{\infty}$ is a reference to a particular Sequence object.

$$ 0, 1, 2, \ldots $$

Is another way to reference the same sequence object.

$a_n = n\forall n\in \mathbb{N}$, $(a_n)_{n=0}^{\infty}$ is yet another way to reference that Sequence object. You get the point.

However, the notation $(\cdot )_{n=0}^{\infty}$, where $\cdot$ is some arbitrary expression involving the symbol $n$, is a sort of template for referencing Sequence objects, like $(\frac{1}{n+1} )_{n=0}^{\infty}$, or something. Thus, I'd like to be able to say:

One way to denote a sequence is $(arb\_expression(n))_{n=0}^{\infty}$. Where arb_expression means you can substitute any "well-defined" expression there involving one dummy symbol $n$. Is there a well-defined mathematical notation for doing so?

EDIT: to be clear, I'm asking about how to define, in a mathematically pleasing way, the way in which Sequence objects are denoted (and of course, extending beyond sequences). I know it can be expressed well enough in words (I just did so), but I'm asking for a method that is "mathematically pleasing," in a sense.

Answer 1

Hint: Note that sequences $(a_n)_{n=0}^\infty$ are just functions with domain $\mathbb{N}$. So, whatever can be said about functions with domain $\mathbb{N}$ can also be said about sequences.

When considering for instance a real-valued sequence $(a_n)_{n=0}^\infty$ we can equivalently consider a function \begin{align*} &f:\mathbb{N}\rightarrow \mathbb{R}\\ &f(n)=a_n \end{align*} and each rule which is mathematically valid can be applied to specify $a_n$.

Answer 2

The notation you provided—i.e., $(a_n)_{n\in\Bbb N}$ or $\{a_n\}_{n\in\Bbb N}$—is perfect. It is shorthand for this:

$$\{a_n : n\in\Bbb N\}$$

The colon is read as “such that” and allows you to provide parameters for your dummy variable. If you have multiple parameters, use $\wedge$ for “and” and/or $\vee$ for “or.”

(You can of course switch out $n\in\Bbb N$ for $n=1$ to $\infty$.)