Is the set of all infinite sequences vector space? 
Let $V$ consist of all sequences $\{a_n\}$ in field $F$ that have only a finite number of nonzero terms $a_n$. If $\{a_n\}$ and $\{b_n\}$ are in $V$ and $t \in F$, define $\{a_n\}+\{b_n\}=\{a_n+b_n\}$ and $t\{a_n\}=\{ta_n\}$. With these operations $V$ is a vector space

Why finite number of nonzero terms is required ?
 A: Summing up the comments, if $W$ is the vector space consisting of all sequences $\{a_n\}$ in a field $F$, then $W$ is a vector space, and $V$ would be a subspace of $W$. The key thing to note is that although $V$ and $W$ are both vector spaces, they are distinct vector spaces.
Some key differences between $V$ and $W$: for $V$ it's really easy to find a basis. For instance, the set of sequences $\{\{a_i=0\,\forall\,i\neq k,a_k=1\}:k\in\mathbb{N}\}$ is a basis of $V$ but not a basis of $W$ (e.g $a = (1,1,1,\ldots)$ is an example of something in $W$ that cannot be written as a finite combination of vectors from the 'basis'). This simplicity helps with proofs.
In general, just a word of warning for reading mathematics: just because an author does things a certain way doesn't mean that that way is required. It's possible that it works in a more general case, and that the author is making the argument simpler and more concrete and easier to understand. Critiquing things in general is good, but just because it's done doesn't necessarily mean it's necessary :)
