The representations of quivers of type A_n. A fast approach

It is well-known that a quiver Q of type A_n is representation-finite, and that its indecomposable representations are thin (all Jordan-Hoelder multiplicities are 0 or 1). By now, various methods of proof are known. The aim of this note is to provide a straight-forward arrangement of possible arguments in order to avoid indices and clumsy inductive considerations, but also avoiding somewhat fancy tools such as the Bernstein-Gelfand-Ponomarev reflection functors or bilinear forms and root systems. The proof we present deals with representations in general, not only finite-dimensional ones. We only use first year linear algebra, namely the existence of bases of vector spaces V, W compatible with a given linear map V -> W, and the existence of a basis of a vector space which is compatible with two given subspaces.