Let $S$ be a nonempty set and $F$ a field. Let $C(S,F)$ denote the set of all functions $f \in G(S,F)$ such that $f(s) = 0$ for all but a finite number of elements of $S$. Prove that $C(S,F)$ is a subspace of $G(S,F)$.
I'm trying to show that $cf \in C(S,F)$ for some $ c \in F$, where $f(s) = 0$ for all but a finite number of elements of $S$. But if $c=0$, we have that $cf=0$ for all elements of $S$. I was wondering how this still fits the property of the subspace $C$ where $f(s)=0$ for all but a finite number of elements of $S$.