# Soft question: $(a_n) \in A$ or $(a_n) \subseteq A$ f0r sequences?

I have always used, in place of the full, unambiguous (but clumsy?) statement namely

"Let $(a_n)_{n\geq 1}$ be a sequence where $a_n \in A$ for $n\geq 1$."

the short version

"Let $(a_n)_{n\geq 1} \in A$."

Although somebody pointed out that this is apparently rather ambiguous and suggested

"Let $(a_n)_{n\geq 1} \subseteq A$."

which is in fact wrong, considering $(a_n)_{n\geq 1}$ is not a set. I believe the shorthand which should be used and is nonetheless still correct is

"Let $(a_n)_{n\geq 1} \in A^{\mathbb{N}}$."

treating the sequence as a function on the naturals. But this is a version (and perhaps the only one) I have never seen anybody using. Which one do you usually prefer? Which ones are acceptable / passable / completely unacceptable to you?

• I prefer $\subseteq$ over $\in$, but usually I avoid both and just write 'Let $(a_n)_n$ be a sequence in $A$'. Jan 1 '14 at 13:27
• @Svinepels Your comment is better than the best answer, so what should I do... Jan 2 '14 at 9:44

The first one is correct.

The second one is most likely incorrect, depending on what $A$ is and it doesn't convey the same as the first one. The same goes for the third one.

The fourth one is correct, but it's a little weird that you refer to the natural numbers in two different ways. I'd use $(a_n)_{n\in \mathbb N}\in A^{\mathbb N}$ instead.

I usually prefer the first one and I have no problem with the fourth one. I abominate the second and third options.

• I disagree about $(a_n)_{n \ge 1} \subseteq A$ - it's fairly commonplace and it's easy to decipher (on the basis that it's highly unlikely to mean anything else). Jan 1 '14 at 13:53
• You can look on a sequence as on structure – its range is universe and the function itself gives structure, in this view $(a_n)_n ⊆ A$ makes a good sense. Jan 1 '14 at 14:02
• @CliveNewstead Would you say $(a_n)_{n\geq 1} \subseteq A$ is easier to decipher than $(a_n)_{n\geq 1} \in A$ then? Jan 1 '14 at 15:40
• @user71815: Yes. For example, suppose $A$ is a set of sequences of natural numbers; then you'd expect $(A_n)_{n \ge 1} \in A$ to refer to an element of this set, i.e. a sequence of natural numbers. But if $\in$ were allowed to refer to sequences too, then this notation might also mean a sequence of sequences of natural numbers! If this notation were allowed, it would be ambiguous. However, $(a_n)_{n \ge 1} \subseteq A$ would, in this setting, unambiguously refer to a sequence of sequences of natural numbers. Jan 1 '14 at 20:57
• @CliveNewstead If that's what you mean, then that's why I typed 'most likely', because usually $A$ isn't a set of sequences. Jan 1 '14 at 21:03

Maybe the best would be to use “let $S: \mathbb{N} \to A$ be a sequence” and then refer to its elements by $S(n)$ and we could write $S \to x$ rather than $\lim_{n \to ∞} S(n) = x$. However it is common to have names of all elements of the sequence (like $a_n$, $x_n$ for $n ∈ \mathbb{N}$) rather than the sequence itself.