I will explain a simple proof of the following statement: K-theory of stable infinity-categories commutes with infinite products. This result is due to Kasprowski and Winges (2018), and their proof uses infinity-version of Grayson's construction of binary acyclic complexes to reduce the statement to K_0. A certain non-trivial argument is needed to reduce the length of these binary acyclic complex (namely, length 7 is sufficient).

Our proof instead uses sheaves on the real line instead of the Grayson's construction. The argument reduces to the well-known elementary statement: the derived category of a Dynkin quiver does not depend on the orientation of arrows.