fortunately you only need to show 3 things true for W:

1) the constant 0-sequence has only finitely many non-zero terms. (this is easy).

2) the sum of any two sequences with only finitely many non-zero terms has only finitely many non-zero terms.

(hint: any sequence

with only finitely many non-zero terms has a maximim non-zero term, say

. if

is another such sequence, it has a maximum non-zero term, say

. show that if t = max{k,m}, all

for n > t are 0).

3) you must verify that the sequence

has only finitely many non-zero terms if

does. this is also easy.