# How do we denote the set of all numbers $a_n$ in a sequence $\{a_n\}$?

I have a question about terms and notation related to sequences.

There is an end-of-chapter problem in Chapter 22 of Spivak's Calculus that says

...prove that the set of all numbers $$a_n$$ actually has a maximum member

How can we denote this set?

$$\{ a_n: a_n \in Image(a) \}$$

seems to me to be correct. Is there some standard way of denoting this set?

Here is some context based on this particular chapter in the book

First, there is the following definition

An infinite sequence of real numbers is a function whose domain is $$\mathbb{N}$$.

He then says

From the point of view of this definition, a sequence should be designated by a single letter like $$a$$, and particular values by

$$a(1),a(2),a(3),...$$

but the subscript notation

$$a_1, a_2, a_3,...$$

is almost always used instead, and a sequence is usually denoted by a symbol like $$\{a_n\}$$. Thus $${n}$$, $$\{(-1)^n\}$$, and $$\{1/n\}$$ denote the sequences $$\alpha$$, $$\beta$$, and $$\gamma$$ defined by

$$\alpha_n=n$$

$$\beta_n=(-1)^n$$

$$\gamma_n=\frac{1}{n}$$

• $\{a_n:n\in \Bbb N\}$. By the way, you should use $(a_n)$, not $\{a_n\}$, to denote a sequence. Oct 17, 2022 at 22:18
• Given the definition, your suggestion looks correct. Though it could just be written as $Image(a)$. Oct 17, 2022 at 22:18
• @ShawSa: I would say that $Image(a)$ would not be immediately understood by most mathematicians (and neither would $\{a_n:a_n\in Image(a)\}$, although this version is clearly inferior to your suggested improvement). Oct 17, 2022 at 22:21
• @TonyK I believe it would if it was understood that $a$ were a function. From the context, I think it's appropriate to use the notation of the book the OP is working out of. In general, I agree with you though. The notation you suggested is probably best (hence the up-vote), unless one wanted to use theorems about images of functions. Oct 17, 2022 at 22:26
• I think the most clear, unambiguous way of denoting the set would be $\{a_n : n \in \Bbb{N}\}$. Oct 17, 2022 at 22:45

$$\{a_n\mid n\in\mathbb{N}\}$$ does the trick. For sequences, you might also encounter the notation $$\{a_1,a_2,a_3,\dots\}$$
If you feel like using a shorter (yet less standard notation), you could write the elements of the sequence as $$a(n)$$ (instead of $$a_n$$). Here you could use $$a(\mathbb{N})$$ or $$\text{Im}(a)$$ to denote the desired set (I would avoid this if you are not representing the elements of $$(a_n)_{n\in\mathbb{N}}$$ as $$a(n)$$).
In general: let $$f\colon X\to Y$$ be a function. Let $$A\subseteq X$$ be a subset. One generally uses the notation $$f(A)=\{f(x)\mid x\in A\}.$$ In case $$A=X$$, we write $$\text{Im}(f)$$.
So if $$f:\Bbb N\to X$$ is such that $$a_n=f(n)$$, then you want $$\rm{im}(f)=f(\Bbb N)$$.