Imprecisions of usual language. What about the following?:
Definition We say that the sequence has finitely many non zero terms iff there exists such that for all .
This is just what all mathematicians interpret with the expression the sequence has finitely many non zero terms.
Question: Has the zero sequence finitely many non zero elements?. You are going to say yes.
I have a sequence which has (only) three non-zero terms. Can the sequence be 0? No
I have a sequence which has no terms* which aren't zero. The sequence could be 0, but nothing else.
I have a sequence which has only no terms which aren't zero. The sequence has to be zero.
So W is a subspace because it contains only 0.
* a zero number of non-zero terms. zero is a finite number. Reference Plato