The second thing you must show is that the set is closed under addition and scalar multiplication. I am assuming "coordinate-wise" addition and scalar multiplication here: and . Scalar multiplication is easy: all but a finite number of the are 0 and any scalar times 0 is 0. So all but a finite number of are 0.
For addition, suppose has n non-zero entries and has m non-zero entries. Can you see that has, at most, m+n non-zero entries?