Proving C(S,F) is a Subspace of F(S,F)

[Riemannliness]

Here's one I've been stewing over:
- Let S be a nonempty set of F, and F a field.
- Let F(S,F) be the set of all functions from S to the field F.
- Let C(S,F) denote the set of all functions f $$\in$$ F(S,F), such that f(s) = 0 for all but a finite number of elements in S (s $$\in$$ S).
Prove that C(S,F) is a subspace of F(S,F).

It's simple to show that the space is closed under addition and scalar multiplication, but I'm having a hard time finding a zero. It certainly isn't the zero function because that function is not nonzero at any finite number of points in S. I've played with a few ideas, but it always comes down to the ambiguity of the definition of S and the specifics of the finite nonzero points mapped by the functions. I can find functions that work for each specific case, but not one that works in all cases. I feel like I'm missing something relatively simple, so hints (BUT NOT ANSWERS) would be appreciated .

 

[Office_Shredder]

Being 0 for all elements of S probably counts as being 0 for all but finitely many elements of S

 

[Riemannliness]

You're probably right. But I was under the impression that it had to be nonzero at least a couple points. Would it be possible if that were the case?

 

[HallsofIvy]

Obviously not, since in that case you would not have a "0" vector.