I'm try to understand the following notation fully:
$(A_n)_{n \in \mathbb{N}}$
Does this notation mean that $A_n$ is an unlimited sequence defined for all n belonging to the natural numbers, meaning for n=1,2,3,4... all the way to infinity? Or does it mean that $A_n$ is a sequence, n belonging to the natural numbers, where $A_n$ could be defined all for n (infinite) or only some.
Hope my question is understandable.
On
If $X$ is any collection (possibly a collection of sets, or not), then the notation $(A_n)_{n\in\Bbb N}$ is used to denote a function $\Bbb N \to X$ such that $1 \mapsto A_1, 2\mapsto A_2,\ldots,$ and so on for each natural number.
Formally, $(A_n)_{n\in\Bbb N}$ is called a sequence of elements of $X$, but it is only a function whose domain is the natural numbers and whose codomain is the set $X$.
If you want to be even more formal, $$ (A_n)_{n\in\Bbb N} = \{(1, A_1),(2,A_2),\ldots\} = \{(n,A_n)\in\Bbb N\times X\}, $$ and this is what we mean when we emphasize that the sequence is itself a function.
The notation $(A_n)_{n \in \mathbb{N}}$ means that $A_n$ is defined for any $n \in \mathbb{N}$, not only parts of $\mathbb{N}$. It denotes indeed a sequence. To phrase it otherwise, for any $n$, there is always something called $A_n$.
I hope this answers your question, for it is not very clear.