Spanning Set Definition
If all the vectors in $V$ can be written as a linear combination of the ones in $S$ then $S$ spans $V$. Your concern about there being some linear combinations of elements in $S$ that are not in $V$ is trivial. Since each element of $S$ is an element of $V$ and $V$ is a vector space, all linear combinations of elements in $S$ will be elements of $V$. This is because of the axioms of a vector space, namely that it is closed under addition and scalar multiplication.