Meaning of $\{a_{n}\}_{n \in \mathbb{N}}$ indexed sequence notation I have doubts about the notation of sequence $\{a_{n}\}_{n \in \mathbb{N}}$. If I have a finite set $A=\{1,2,3,4,5\}$, does the index sequence notation describe a single possible sequence $\{a_{n}\}_{n \in \mathbb{N}}=(1,2,3,4,5)$? Or does this notation describe multiple possible sequences?
 A: The notation $\{a_n\}_{n \in \mathbb{N}}$ denotes an infinite sequence $(a_1, a_2, a_3, ...)$, where you have an element $a_n$ for every $n \in \mathbb{N}$. If you're considering this to be a sequence in $A = \{1,2,3,4,5\}$ (i.e. for every $n \in \mathbb{N}$, we have $a_n \in A$), then your sequence $\{a_n\}_{n \in \mathbb{N}}$ could be all sorts of things: $(1,2,1,2,1,2,...)$, $(4,4,4,4,4,4,...)$, $(3,1,4,1,5,2,5,...)$. Importantly, these sequences are all infinite, and are not influenced by the structure of the set $A$.
Technically, a sequence $\{a_n\}_{n \in \mathbb{N}}$ whose elements are in $A$ is formally defined to be a function $f : \mathbb{N} \rightarrow A$. The element $a_n$ is the image of $n$ under $f$; that is, $a_n = f(n)$.
My personal preference is actually to not use the notation $\{a_n\}_{n \in \mathbb{N}}$ and instead use the notation $(a_n)_{n \in \mathbb{N}}$, to emphasize that this is not a set: there is an order, and you can repeat elements. You might see people write $\{a_n\}_{n \in \mathbb{N}} \subseteq A$ to indicate that the elements of the sequence are in $A$; this is technically an abuse of notation.
