Clarification of sequence space definition Let $(x_n)$ denote a sequence whose $n$th term is $x_n$, and $\{x_n\,:\,n\in\mathbb{N}\}$ denote the set of all elements of the sequence.  I have a text that states

Note that $\{x_n\,:\,n\in\mathbb{N}\}$ can be a finite set even though $(x_n)$ is an infinite sequence.

To me this seems to be a contradiction.  Can any reiterate this quotation to shed light on its meaning?  I am confused on the difference between $(x_n)$ and $x_n$, and $\{x_n\,:\,n\in\mathbb{N}\}$.  Thanks all.

If $x_n=1$ for all $n$ then $\{x_n:n\in \mathbb{N}\}=\{1\}$. So the comment is really saying that set notation loses several pieces of information from the sequence. Specifically you lose information on the order of elements and also the number of times a specific element appears in the sequence.
By way of comparison, you might like to see the definition of a multiset which, formally, is a function $f\colon X\rightarrow \mathbb{N}\setminus\{0\}$ which assigns to each element of the set $X$ the number of times it appears in the multiset. Informally it just means that we allow repitition of elements in our multiset and we don't forget how many times an element appears. Importantly though, we do not consider any kind of ordering information in the multiset, so $\{a,a,a,b,c,c\}=\{a,b,c,a,c,a\}$ as multisets.
If we extend the definition of multiset to include elements with possibly infinite values*, then there is a well defined multiset associated to each sequence, and any such multiset has an infinite number of elements (this is a slight abuse of notation, given that a multiset is formally defined to be a function, but it can be made rigorous).
*That is, we define a multiset to be a function $X\rightarrow (\mathbb{N}\setminus\{0\})\cup\{\infty\}$
