Title: Shortening binary complexes and commutativity of $K$-theory with infinite products,
Abstract: We show that in Grayson's model of higher algebraic $K$-theory using binary
acyclic complexes, the complexes of length two suffice to generate the whole
group. Moreover, we prove that the comparison map from Nenashev's model for
$K_1$ to Grayson's model for $K_1$ is an isomorphism. It follows that algebraic
$K$-theory of exact categories commutes with infinite products.