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 [tex]\in[/tex] F(S,F), such that f(s) = 0 for all but a finite number of elements in S (s [tex]\in[/tex] 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 .