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?