If we don't insist that all but finitely many entries are non-zero, then we get the (set-theoretic) product of countably many copies of ${\mathbb R}$. If we look at the degree-n Grassmanian of this object, then this is not the object defined in Milnor Stasheff. Hence there exists at least one theorem that is true for the object they define, which is not true for the object we have just defined.