# Proof of vector subspace for a set of finite functions

Here is what to prove:

$$Fun ^ {end} (M, K): = {f: M rightarrow K | f (m) neq 0$$ only for a finite number of $$m in M ​​}$$ is a subspace of $$Amusement (M, K)$$

here is the plan

1. CA watch $$Fun ^ {end} (M, K) neq emptyset$$
2. CA watch $$Fun ^ {end} (M, K)$$ is closed under addition
3. CA watch $$Fun ^ {end} (M, K)$$ is closed under scalarm multiplication

Well, I'll be honest, I do not know how to start with the first point. Maybe we can assume that there is $$infinite$$ $$m$$ such as $$f (m) = 0$$ and give a counterexample to show that the hypothesis is false so $$Fun ^ {end} (M, K) neq emptyset$$ must be true?
But how could I continue then?