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.

A sequence of real numbers is a function, not a set. Thus, for instance, the sequence $(x_n)$ is actually a function $f:\mathbb N \to \mathbb R$, where we have the equality $x_n =f(n)$. Now, the image of the function is the set $\{x_n\mid n\in \mathbb N\}$, which is a very different thing. An example where this associated set is infinite is for the sequence $(x_n)$ where $x_n=1$ for all $n$. Then the associated set is just $\{1\}$. 
